THE 


[JMBER-SYSTEM  OF  ALGEBRA 


TREATED  THEORETICALLY  AND  HISTORICALLY 


BY 
HENRY   B.   FINE,  Ph.D., 

Professor  of  Mathematics  in  Princeton  College. 


LEACH,  SHEWELL,  &  SANBORN, 
BOSTON  AND  NEW  YORK. 


Copyright,  1890, 
By  HENRY  B.  FINE. 


Typography  by  J.  S.  Gushing  &  Co.,  Boston. 
Presswork  by  Berwick  &  Smith,  Boston. 


PREFACE. 


The  theoretical  part  of  this  little  book  is  an  elementary 
exposition  of  the  nature  of  the  number  concept,  of  the  posi- 
tive integer,  and  of  the  four  artificial  forms  of  number 
which,  with  the  positive  integer,  constitute  the  "number- 
system  "  of  algebra,  viz.  the  negative,  the  fraction,  the  irra- 
tional, and  the  imaginary.  The  discussion  of  the  artificial 
numbers  follows,  in  general,  the  same  lines  as  my  pam- 
phlet :  On  the  Forms  of  Number  arising  in  Common 
'Algebra,  but  it  is  much  moro  exhaustive  and  thorough- 
going. The  point  of  view  is  the  one  first  suggested  by 
Peacock  and  Gregory,  and  accepted  by  mathematicians  gen- 
erally since  the  discovery  of  quaternions  and  the  Ausdeh- 
nungslehre  of  Grassmann,  that  algebra  is  completely  defined 
formally  by  the  laws  of  combination  to  which  its  funda- 
mental operations  are  subject ;  that,  speaking  generally, 
these  laws  alone  define  the  operations,  and  the  operations 
the  various  artificial  numbers,  as  their  formal  or  symbolic 
results.  This  doctrine  was  fully  'developed  for  the  negative, 
the  fraction,  and  the  imaginary  by  Hankel,  in  his  Complexe 
Zahlensystemen,  in  1867,  and  made  complete  by  Cantor's 
beautiful  theory  of  the  irrational  in  1871,  but  it  has  not 
as  yet  received  adequate  treatment  in  English. 

Any  large  degree  of  originality  in  work  of  this  kind  is 
naturally  out  of  the  question.      I  have  borrowed  from  a 

iii 

i  rvooo. 


IV  PREFACE. 

great  many  sources,  especially  from  Peacock,  G-rassmann 
Hankel,  Weierstrass,  Cantor,  and  Thomae  (Theorie  dei 
andlytischen  Functional  einer  complexen  Verdnderliclien) .  1 
may  mention,  however,  as  more  or  less  distinctive  features 
of  my  discussion,  the  treatment  of  number  and  counting 
(§§  1-5)  and  the  equation  (§§4,  12),  the  prominence  given 
the  laws  of  the  determinateness  of  subtraction  and  division, 
and  the  demonstration  of  the  one-to-one  correspondence  be- 
tween numbers  defined  by  regular  sequences  and  the  points 
of- a  line  (§40). 

Much  care  and  labor  have  been  expended  on  the  his- 
torical chapters  of  the  book.  These  were  meant  at  the  out- 
set to  contain  only  a  brief  account  of  the  origin  and  history 
of  the  artificial  numbers.  But  I  could  not  bring  myself  tc 
ignore  primitive  counting  and  the  development  of  numeral 
notation,  and  I  soon  found  that  a  clear  and  connected 
account  of  the  origin  of  the  negative  and  imaginary  i 
possible  only  when  embodied  in  a  sketch  of  the  early  his 
tory  of  the  equation.  I  have  thus  been  led  to  write  i 
resume  of  the  history  of  the  most  important  parts  of  ele- 
mentary arithmetic  and  algebra. 

Moritz  Cantor's  Vorlesungen  uber  die  Oeschichte  der  Mathe 
matik,  Vol.  I,  has  been  my  principal  authority  for  tin 
entire  period  which  it  covers,  i.e.  to  1200  a.d.  For  tin 
little  I  have  to  say  on  the  period  1200  to  1600,  I  hav< 
depended  chiefly,  though  by  no  means  absolutely,  or 
Hankel:  Zur  QescMchte  der  Mathematik  in  Altertum  unci 
Mittelalter.  The  remainder  of  my  sketch  is  for  the  mosi 
part  based  on  the  original  sources. 

HENRY  B.   FINE. 

Princeton,  April,  1891. 


CONTENTS. 

I.    THEORETICAL. 
I.    The  Positive  Integer. 

PAGE 

The  number  concept 3 

Numerical  equality 3 

Numeral  symbols 4 

The  numerical  equation 6 

Counting • 5 

Addition  and  its  laws 6 

Multiplication  and  its  laws * 7 

II.    Subtraction  and  the  Negative  Integer. 

Numerical  subtraction . 8 

Determinateness  of  numerical  subtraction 9 

Formal  rules  of  subtraction 9 

Limitations  of  numerical  subtraction 11 

Symbolic  equations' 11 

Principle  of  permanence.     Symbolic  subtraction 12 

Zero 13 

The  negative 14 

Recapitulation  of  the  argument  of  the  chapter 16 

III.   Division  and  the  Fraction. 

Numerical  division .18 

Determinateness  of  numerical  division 18 

Formal  rules  of  division 19 

Limitations  of  numerical  division 20 

Symbolic  division.     The  fraction '.  21 

Negative  fractions 22 

General  test  of  the  equality  or  inequality  of  fractions 22 

Indeterminateness  of  division  by  zero 23 

v 


Vi  CONTENTS. 

PAGE 

Determinateness  of  symbolic  division 23 

The  vanishing  of  a  product 24 

The  system  of  rational  numbers . . . 25 

IV.    The  Irrational. 

Inadequateness  of  the  system  of  rational  numbers 26 

Numbers  denned  by  "  regular  sequences."     The  irrational 27 

Generalized  definitions  of  zero,  positive,  negative ; 2{T 

Of  the  four  fundamental  operations 29 

Of  equality  and  greater  and  lesser  inequality 31 

The  number  denned  by  a  regular  sequence  its  limiting  value 31 

Division  by  zero 33 

The  number-system  defined  by  regular  sequences  of  rationals  a 

closed  system , .. 34 

V.   The  Imaginary.    Complex  Numbers. 

The  pure  imaginary fc 35 

Complex  numbers , 36 

The  fundamental  operations  on  complex  numbers . . _ . . . 37 

Numerical  comparison  of  complex  numbers 38 

Adequateness  of  the  system  of  complex  numbers. ., 39 

Fundamental  characteristics  of  the  algebra  of  number 39 

VI.   Graphical  Eepresentatton  of  Numbers.     The  Variable. 

Correspondence  between  the  real  number-system  and  the  points 

of  a  line .  • \ 41 

The  real  number-system  "  continuous  "" 43 

The  variable 43,  45 

Correspondence  between  the   complex  number-system   and  the 

.points  of  a  plane 44 

Definitions  of  modulus  and  argument  of  a  complex  number  and  of 

sine,  cosine,  and  circular  measure  of  an  angle 45,  48 

Demonstration  that  a  +  ib  —  p  (cos  0  +  i  sin  6)  =  pe^ 45,  48 

Construction  of  the  points  which  represent  the  sum,  difference, 

product,  and  quotient  of  two  complex  numbers 46,  47 

VII.    The  Fundamental  Theorem  of  Algebra. 

Definitions  of  the  algebraic  equation  and  its  roots 50 

Demonstration  that  an  algebraic  equation  of  the  nth  degree  has 
n  roots 51,  53 


CONTENTS.  Vll 

VIII.    Numbers  Defined  by  Infinite  Series. 

page 

I.  Heal  Series. 

Definitions  of  sum,  convergence,  and  divergence 54 

General  test  of  convergence 55 

Absolute  and  conditional  convergence 55,  57 

Special  tests  of  convergence 57 

Limits  of  convergence 59 

The  fundamental  operations  on  infinite  series 61 

II.  Complex  Series. 

General  test  of  convergence 62 

Absolute  and  conditional  convergence 62 

The  region  of  convergence 63 

A  theorem  respecting  complex  series 64 

The  fundamental  operations  on  complex  series 65 

IX.   The  Exponential  and  Logarithmic  Functions. 

UNDETERMINED   COEFFICIENTS.       INVOLUTION   AND   EVOLUTION.       THE 
BINOMIAL    THEOREM. 

Definition  of  function 66 

Functional  equation  of  the  exponential  function 66 

Undetermined  coefficients 67 

The  exponential  function 68 

The  functions  sine  and  cosine 71 

Periodicity  of  these  functions 72 

The  logarithmic  function 72 

Indeterminateness  of  logarithms 75 

Permanence  of  the  laws  of  exponents 76 

Permanence  of  the  laws  of  logarithms 77 

Involution  and  evolution 77 

The  binomial  theorem  for  complex  exponents 77 


II.    HISTORICAL. 


I.    Primitive  Numerals. 


Gesture  symbols 81 

Spoken  symbols 82 

Written  symbols '. . .  .♦ 84 


Vlll     -  CONTENTS. 

II.    Historic  Systems  of  Notation. 

page 

Egyptian  and  Phoenician 84 

Greek 84 

Roman 85 

Indo- Arabic 86 

III.    The  Fraction. 

Primitive  fractions 90 

Roman  fractions 91 

Egyptian  (the  Book  of  Ahmes) 91 

Babylonian  or  sexagesimal 92 

Greek 93 

IV.    Origin  of  the  Irrational. 

Discovery  of  irrational  lines.     Pythagoras 94 

Consequences  of  this  discovery  in  Greek  mathematics 96 

Greek  approximate  values  of  irrationals 98 

V.    Origin  of  the  Negative  and  the  Imaginary. 

THE    EQUATION. 

The  equation  in  Egyptian  mathematics 99 

In  the  earlier  Greek  mathematics 99 

Hero  of  Alexandria 100 

Diophantus  of  Alexandria 10-1 

The  Indian  mathematics.     Aryabhatta,  Brahmagupta,  Bhaskara  103 

Its  algebraic  symbolism 104 

Its  invention  of  the  negative 105 

Its  use  of  zero 105 

Its  use  of  irrational  numbers 106 

Its  treatment  of  determinate  and  indeterminate  equations 106 

The  Arabian  mathematics.    Alkhwarizmi,  Alkarchi,  Alchayyami  107 

Arabian  algebra  Greek  rather  than  Indian 110 

Mathematics  in  Europe  before  the  twelfth  century Ill 

Gerbe-J .  Ill 

Entra.^e  of  the  Arabian  mathematics.     Leonardo 112 

Mathematics  during  the  age  of  Scholasticism 113 

The  Renaissance.    Solution  of  the  cubic  and  biquadratic  equations  114 
The  negative  in  the  algebra  of  this  period.    Eirst  appearance  of 

the  imaginary 115 

Algebraic  symbolism.     Vieta  and  Harriot 116 

The  fundamental  theorem  of  algebra.     Harriot  and  Girard 117 


CONTENTS.     -  ix 

VI.    Acceptance    of    the    Negative,    the    General   Irrational, 
and  the  Imaginary  as  Numbers. 

PAGE 

Descartes'  Geometrie  and  the  negative 118 

Descartes'  geometric  algebra 119 

The  continuous  variable.    Newton.     Euler 121 

The  general  irrational 121 

The  imaginary,  a  recognized  analytical  instrument 122 

Argand's  geometric  representation  of  the  imaginary 122 

Gauss.     The  complex  number 123 

VII.    Recognition  of  the  Purely  Symbolic  Character  of 
Algebra. 

quaternions.     the  ausdehnungslehre. 

The  principle  of  permanence.    Peacock 124 

The  fundamental    laws    of    algebra.     "  Symbolical    algebras." 

Gregory 126 

Hamilton's  quaternions 128 

Grassmann's  Ausdehnungslehre 129 

The  fully  developed  doctrine  of  the  artificial  forms  of  number. 

Hankel.     Weierstrass.     G.  Cantor 130 

Recent  literature 131 


PRINCIPAL  FOOTNOTES. 

Instances  of  quinary  and  vigesimal  systems  of  notation 82 

Instances  of  digit  numerals 83 

Summary  of  the  history  of  Greek  mathematics 95 

Old  Greek  demonstration  that  the  side  and  diagonal  of  a  square 

are  incommensurable 96 

Greek  methods  of  approximation .  .  98 

Diophantine  equations ...  102 

Alchayy ami's  method  of  solving  cubics  by  the  intersections . 

conies 109 

Jordanus  Nemorarius 113 

The  Summa  of  Luca  Pacioli ,......,..  113 

Regiomontanus . . C ............ . 114 

Algebraic  symbolism , 113,  116 

The  irrationality  of  e  and  7r.    Lindemann ... .......... 70 


I. 

THEORETICAL. 


I.    THE  POSITIVE   INTEGEK, 

AND   THE    LAWS    WHICH    REGULATE   THE   ADDITION    AND 
MULTIPLICATION    OF    POSITIVE    INTEGERS. 

1.  Number.  Separateness  or  distinctness  is  a  primary 
cognition,  being  necessary  even  to  the  cognition  of  things 
as  individuals,  as  distinct  from  other  things. 

The  notion  of  number  is  based  immediately  on  this  pri- 
mary cognition. 

Number  is  that  property  of  a  group  of  distinct  things 
which  remains  unchanged  during  .any  change  to  which  the 
group  may  be  subjected  which  does  not  destroy  the  distmct- 
ness  of  the  individual  things. 

Such  changes  are  changes  of  the  characteristics  of  the 
individual  things  or  of  their  arrangement ;  for  these  do  not 
cause  one  thing  to  split  up  into  more  than  one,  nor  more 
than  one  to  merge  in  one. 

This  characteristic  of  number  may  be  stated  in  a  theorem 
which  is  the  fundamental  postulate  of  arithmetic  : 

The  number  of  things  in  any  group  of  distinct  things  is, 
independent  of  the  characters  of  these  things,  of  the  order  in  ^ 
which  they  may  be  arranged  in  the  group,  and  of  the  manner 
in  which  they  may  be  associated  with  one  another  in  smaller 
groups. 

2.  Numerical  Equality.  The  number  of  things  in  any 
two  groups  of  distinct  things  is  the  same,  when  for  each 
thing  in  the  first  group  there  is  one  in  the  second,  and 
reciprocally,  for  each  thing  in  the  second  group,  one  in  the 
first. 

Thus,  the  :  aber  of  letters  in  the  two  groups,  A,  B,  (7; 
D,  E,  F,  is  cxxc  sarre.     In  Jie  second  group  there  is  a  letter 

I 


4  NUMBER-SYSTEM  OF  ALGEBRA. 

which  may  be  assigned  to  each  of  the  letters  in  the  first : 
as  D  to  A,  E  to  B,  F  to  C\  and  reciprocally,  a  letter  in  the 
first  which  may  be  assigned  to  each  in  the  second :  as  A  to 
D,  B  to  E,  C  to  F. 

Two  groups  thus  related  are  said  to  be  in  one-to-one  (1-1) 
correspondence. 

Underlying  the  statement  just  made  is  the  assumption 
that  if  the  two  groups  correspond  in  the  manner  described 
for  one  order  of  the  things  in  each/ they  will  correspond  if 
the  things  be  taken  in  any  other  order  also ;  thus,  in  the 
example  given,  that  if  E  instead  of  D  be  assigned  to.  A, 
there  will  again  be  a  letter  in  the  group  D,  E,  F,  viz.  D  or 
F,  for  each  of  the  remaining  letters  B  and  C,  and  recipro- 
cally.    This  is  an  immediate  consequence  of  §  1. 

The  number  of  things  in  the  first  group  is  greater  than 
that  in  the  second,  or  the  number  of  things  in  the  second 
less  than  that  in  the  first,  when  there  is  one  thing  in  the 
first  group  for  each  thing  in  the  second,  but  not  reciprocally 
one  in  the  second  for  each  in  the  first. 

3.  Numeral  Symbols.  As  regards  the  number  of  things 
which  it  contains,  therefore,  a  group  may  be  represented 
by  any  other  group,  e.g.  of  the  fingers  or  of  simple  marks, 
I's,  which  stands  to  it  in  the  relation  of  correspondence 
described  in  §  2.  This  is  the  primitive  method  of  repre- 
senting the  number  of  things  in  a  group  and,  like  the 
modern  method,  makes  it  possible  to  compare  numerically 
groups  which  are  separated  in  time  or  space. 

The  modern  method  of  representing  the  number  of  things 
in  a  group  differs  from  the  primitive  only  in  the  substitu- 
tion of  symbols,  as  1,  2,  3,  etc.,  or  numeral  words,  as  one, 
two,  three,  etc.,  for  the  various  groups  of  marks  I,  II,  III, 
etc.  These  symbols  are  the  positive  integers  of  arith- 
metic. 

A  positive  integer  is  a  symbol  for  the  number  of  things  in  a 
group  of  distinct  things. 


THE    POSITIVE    INTEGER.  5 

For  convenience  we  shall  call  the  positive  integer  which 
represents  the  nnmber  of  things  in  any  group  its  numeral 
symbol,  or  when  not  likely  to  cause  confusion,  its  number 
simply,— this  being,  in  fact,  the  primary  use  of  the  word 
"  number  "  in  arithmetic. 

In  thj3  following  discussion,  for  the  sake  of  giving  our 
statements  a  general  form,  we  shall  represent  these  numeral 
symbols  by  letters,  a,  b,  c,  etc. 

4.  The  Equation.  The  numeral  symbols  of  two  groups 
being  a  and  b ;  when  the  number  of  things  in  the  groups  is 
the  same,  this  relation  is  expressed  by  the  equation 

a  =  b; 

when  the  first  group  is  greater  than  the  second,  by  the 
inequality 

a>b; 

when  the  first  group  is  less  than  the  second,  by  the  ine- 
quality 

a<b. 

A /numerical  equation  is  thus  a  declaration  in  terms  of  the 
numeral  symbols  of  two  groups  and  the  symbol  =  that  these 
groups  are  in  one-to-one  correspondence  (§2). 

5.  Counting.  The  fundamental  operation  of  arithmetic 
is  counting. 

To  count  a  group  is  to  set  up  a  one-to-one  correspondence 
between  the  individuals  of  this  group  and  the  individuals 
of  some  representative  group. 

Counting  leads  to  an  expression  for  the  number  of  things 
in  any  group  in  terms  of  the  representative  group  :  if  the 
representative  group  be  the  fingers,  to  a  group  of  fingers ; 
if  marks,  to  a  group  of  marks ;  if  the  numeral  words  or 
symbols  in  common  use,  to  one  of  these  words  or  symbols. 

There  is  a  difference  between  counting  with  numeral 
words  and  the  earlier  methods  of  counting,  due  to  the  fact 


6  NUMBER-SYSTEM  OF  ALGEBBA. 

that  the  numeral  words  have  a  certain  recognized  order. 
As  in  finger-counting  one  ringer  is  attached  to  each  thing 
counted,  so  here  one  word ;  but  that  ivord  represents  numer- 
ically not  the  thing  to  which  it  is  attached,  but  the  entire 
group  of  which  this  is  the  last.  The  same  sort  of  counting 
may  be  done  on  the  fingers  when  there  is  an  agreement  as 
to  the  order  in  which  the  fingers  are  to  be  used;  thus  if 
it  were  understood  that  the  fingers  were  always  to  be  taken 
in  normal  order  from  thumb  to  little  finger,  the  little  finger 
would  be  as  good  a  symbol  for  5  as  the  entire  hand. 

6.  Addition.  If  two  or  more  groups  of  things  be  brought 
together  so  as  to  form  a  single  group,  the  numeral  symbol 
of  this  group  is  called  the  sum  of  the  numbers  of  the  sepa- 
rate groups. 

If  the  sum  be  s,  and  the  numbers  of  the  separate  groups 
a,  b,  c,  etc.,  respectively,  the  relation  between  them  is  sym- 
bolically expressed  by  the  equation 

s  =  a-\-b  +  c  +  etc., 

where  the  sum-group  is  supposed  to.  be  formed  by  joining 
the  second  group  —  tp  which  b  belongs  —  to  the  first,  the 
third  group  —  to  which  c  belongs  —  to  the  resulting  group, 
and  so  on. 

The  operation  of  finding  s  when  a,  b,  c,  etc.,  are  known,  is 
addition. 

Addition  is  abbreviated  counting. 

Addition  is  subject  to~th!TW~oHfollowiiig  laws,  called  the 
commutative  and  associative  laws  respectively,  viz. : 

I.    a-hb  =  b  +  a. 
II.    a  +  (5  +  c)=(a-f&)f  a 
Or, 

I.    To  add  b  to  a  is  the  same  as  to  add  a  to  b. 
II.    To  add  the  sum  of  b  and  c  to  a  is  the  same  as  to  add 
c  to  the  sum  of  a  and  b. 


THE  POSITIVE  INTEGER.  7 

Both  these  laws  are  immediate  consequences  of  the  fact 
that  the  sum-group  will  consist  of  the  same  individual 
things,  and  the  number  of  things  in  it  therefore  be  the 
same,  whatever  the  order  or  the  combinations  in  which  the 
separate  groups  are  brought  together  (§1). 

7.  Multiplication.  The  sum  of  b  numbers  each  of  which 
is  a  is  called  the  product  of  a  by  b,  and  is  written  a  x  b,  or 
a  •  b,  or  simply  ab. 

The  operation  by  which  the  product  of  a  by  &  is  found, 
when  a  and  &.are  known,  is  called  multiplication. 

Multiplication  is  an  abbreviated  addition.. 

Multiplication  is  subject  to  the  three  following  laws, 
called  respectively  the  commutative,  associative,  and  distrib- 
utive laws  for  multiplication,  viz. : 

III.  ab  =  ba. 

IV.  a(bc)  =($$.■*> 

V.   a(b  +  c)  =  ab-\-ac. 

Or, 

III.  The  product  of  a  by  b  is  the  same  as  the  product  of 
b  by  a. 

TV.  The  product  of  a  by  be  is  the  same  as  the  product  of 
ab  by  c. 

V.    The  product  of  a  by  the  sum  of  b  and  c  is  the  same 
as  the  sum  of  the  product  of  a  by  b  and  of  a  by  c. 

These  laws  are  consequences  of  the  commutative  and 
associative  laws  for  addition.     Thus, 

III.  The  Commutative  Law.  The  units  of  the  group  which 
corresponds  to  the  sum  of  b  numbers  each  equal  to  a  may 
be  arranged  in  b  rows  containing  a  units  each.  But  in  such 
an  arrangement  there  are  a  columns  containing  ft  units  each ; 
so  that  if  this  same  set  of  units  be  grouped  by  columns 
instead  of  rows,  the  sum  becomes  that  of  a  numbers  each 


8  NUMBETt-SYSTEM  OF  ALGEBRA. 

equal  to  b,  or  ba.     Therefore  ab  =  ba,  by  the  commutative 
and  associative  laws  for  addition. 

IV.  The  Associative  Law. 

^b]c  =  c  sums  such  as  (a-fa-i to  b  terms) 

=  a  +  a  +  aH to  be  terms  (by  the  associative 

law  for  addition) 
=  a(bc). 

V.  The  Distributive  Law. 

a(b  +  c)  =  a  +  a  -f-  a  -f  •  •  •  to  (b  -f-  c)  terms 

=  (a  +  cH to  &  terms) -|-(a  +  aH tp  c  terms) 

(by  the  associative  law  for  addition), 
=  ab  +  a  c. 

The  commutative,  associative,  and  distributive  laws  for 
sums  of  any  number  of  terms  and  products  of  any  number 
of  factors  follow  immediately  from  I-V.  Thus  the  product 
of  the  factors  a,  b,  c,  d,  taken  in  any  two  orders,  is  the  same, 
since  the  one  order  can  be  transformed  into  the  other  by 
successive  interchanges  of  consecutive  letters. 


II.  SUBTRACTION  AND  THE  NEGATIVE  INTEGER. 

8.  Numerical  Subtraction.  Corresponding  Tjq^very,math- 
ematical  operation  there  is  another,  commonly  called  its  in- 
verse, which  exactly  undoes  what  the  operation  itseff  does. 
Subtraction  stands  in  this  relation  to  addition,  and  division 
to  multiplication. 

To  subtract  b  from  a  is  to  find  a  number  to  which  if  b  be 
added,  the  sum  will  be  a.  The  result  is  written  a  —  b ;  by 
definition,  it  identically  satisfies  the  equation 

VI.  (a-b)  +  b  =  a-, 


SUBTRACTION.  9 


that  is  to  say,  a  —  b  is  the  number  belonging  to  the  group 
which  with  the  6-group  makes  up  the  a-group. 

Obviously  subtraction  is  always  possible  when  b  is  less 
than  a,  but  then  only.  Unlike  addition,  in  each  application 
of  this  operation  regard  must  be  had  to  the  relative  size  of 
the  two  numbers  concerned. 

9.  Determinateness  of  Numerical  Subtraction.  Subtrac- 
tion, when  possible,  is  a  determinate  operation.  There  is 
but  one  number  which  will  satisfy  the  equation  x  4-  b  =  a, 
but  one  number  the  sum  of  which  and  b  is  a.  ui  other 
words,  a  —  b  is  one-valued. 

For  if  c  and  d  both  satisfy  the  equation  x  +  b  =  a,  since 
then  c-\-b  =  a  and  d  +  b  =  a,  c  +  b  =  d-\-b\  that  is,  a  one-to- 
one  correspondence  may  be  set  up  between  the  individuals 
of  the  (c  +  b)  ajid  {d  -f-  b)  groups  (§4).  The  same  sort  of 
correspondence,  however,  exists  between  any  b  individuals 
of  the  first  group  and  any  b  individuals  of  fche  secon|Lk it 
must,  therefore,  exist  between  the  remaining  c  of  the  fltrst 
and  the  remaining:  d  of  the  second,  or  c  =  d. 

This  characteristic^of  subtraction  is  of  the  same  order  of 
importance  as  the  commutative  and  associative  laws,  and 
we  shall  add  to  the  group  of  laws  I-V  and  definition  VI  — 
as  being,  like  them,  a  fundamental  principle  in  the  follow- 
ing discussion  —  the  theorem 

(  a  =  b, 

which  may  also  be  stated  in  the  form  :  If  one  term  of  a  sum 
changes  while  the  other  remains  constant,  the  sum  changes. 
The  same  reasoning  proves,  also,  that 

(  As  a  +  c  >  or  <  b  +  c, 
1  a  >  or  <  b. 

10.  Formal  Rules  of  Subtraction.  All  the  rules  of  sub- 
traction are  purely  formal  consequences  of  the  fundamental 


10  NUMBER-SYSTEM  OF  ALGEBRA. 

laws  I-V,  VII,  and  definition  VI.  They  must  follow,  what- 
ever the  meaning  of  the  symbols  a,  b,  c,  +,  — >  = ;  a  fact 
which  has  an  important  bearing  on  the  following  discussion. 
It  will  be  sufficient  to  consider  the  equations  which  fol- 
low. For,  properly  combined,  they  determine  the  result  of 
any  series  of  subtractions  or  of  any  complex  operation  made 
up  of  additions,  subtractions,  and  multiplications. 

1.  a  —  (b  +  c)  =  a  —  b  —  c  =  a  —  c  —  b. 

2.  a  —  (b  —  c)  =  a  —  b-\-c. 

3.  a-\-b—  b  =  a. 

4.  a  +  (&  —  c)  =  a  +  b  —  c  =  a  —  c  +  5. 

5.  a  (b  —  c)  =  ab  —  ac. 

For  1.  a  —  b  —  c  is  the  form  to  which  if  first  c  and  then  b 
be  added;  or,  what  is  the  same  thing  (by  I), 
first  b  and  then  c ;  or,  what  is  again  the  same 
thing  (by  II)/5-}-c  at  once, — the  sum  pro- 
duced is  a  (by  VI)r~a  —  b  —  c  is  therefore  the 
same  as  a  —  c  —  b,  which  is  as  it  stands  the 
form  to  which  if  b,  then  c,  be  added  the  sum  is 
a;  also  the  same  as  a—  (6  +  c),  which  is  the 
form  to  which  if  b  -+-  c  be  added  the  sum  is  a. 


2. 

a  —  (b  —  c)  =  a  —  (b  —  c)  —  c  -f  c, 

Def.  VI. 

=  a  —  (b  —  c  -f-  c)  +  c, 

Eq.  1. 

=  a  ■—  b  +  e. 

Def.  VI. 

3. 

a  +  b  —  b  +  b  =  a  -f-  fc. 

But       a  -f  6  =  a  +  6. 

Def.  VI. 

.-.  a -j- 6  —  &  =  a. 

Law  VII. 

4. 

a  +  &  —  c  =  a  +  (&  —  c  -f  c)  —  c, 

Def.  VI. 

.^     =a  +  (6  — c). 

Law  II,  Eq.  3. 

5. 

a&  —  ac  =  a  (6  —  c  +  c)  —  ac, 

Def.  VI. 

=  a  (b  —  c)  +  ac  —  ac, 

Law  V. 

=  a(6—c). 

Eq.  3. 

SUBTRACTION.  11 

Equation  3  is  particularly  interesting  in  that  it  defines 
addition  as  the  inverse  of  subtraction.  Equation  1  declares 
that  two  consecutive  subtractions  may  change  places,  are 
commutative.  Equations  1,  2,  4  together  supplement  law 
II,  constituting  with  it  a  complete  associative  law  of  addi- 
tion and  subtraction ;  and  equation  5  in  like  manner  supple- 
ments law  V. 

11.  Limitations  of  Numerical  Subtraction.  Judged  by 
the  equations  1-5,  subtraction  is  the  exact  counterpart  of 
addition.  It  conforms  to  the  same  general  laws  as  that 
operation,  and  the  two  could  with  fairness  be  made  to 
interchange  their  rdles  of  direct  and  inverse  operation. 

But  this  apparent  equality  vanishes  when  the  attempt  is 
made  to  interpret  these  equations.  The  requirement  that 
subtrahend  be  less  than  minuend  then  asserts  itself  as  a 
fatal  limitation.  It  makes  the  range  of  subtraction  much 
narrower  than  that  of  addition.  It  renders  the  equations 
1-5  available,  for  special  classes  of  values  of  a,  b,  c  only. 
If  it  must  be  insisted  on,  even  so  simple  an  inference  as 
that  a  —  (a  -f- 5)  -f-2  b  is  equal  to  b  cannot  be  drawn,  and 
the  use  of  subtraction  in  any  reckoning  with  symbols  whose 
relative  values  are  not  at  all  times  known  must  be  pro- 
nounced unwarranted. 

One  is  thus  led  perforce  to  ask  whether  int.erpretability 
•is  after  all  necessary  to  the  validity  of  reckonings  and,  if 
not,  to  seek  to  free  subtraction  and  the  rules  of  reckoning 
with  the  results  of  subtraction  from  this  crippling  limi- 
tation. 

12.  Symbolic  Equations.  Principle  of  Permanence.  Sym- 
bolic Subtraction.  In  pursuance  of  this  inquiry  one  turns 
first  to  the  equation  (a  —  b)  +  b  —  a,  which  serves  as  a 
definition  of  subtraction  when,  b  is  less  than  a. 

This  is  an  equation  in  the  primary  sense  (§4)  only  when 
a  —  b  is  a  number.     But  in  the  broader  sense,  that 


12  NUMBER-SYSTEM  OF  ALGEBRA. 

An  equation  is  any  declaration  of  the  equivalence  of  definite 
combinations  of  symbols  —  equivalence  in  the  sense  that  one 
may  be  substituted  for  the  other,  — 

(a  —  b)-\-b  =  a  may  be  an  equation,  whatever  the  values 
of  a  and  b. 

And  if  no  different  meaning  has  been  attached  to  a  —  b, 
and  it  is  declared  that  a  —  b  is  the  symbol  which  associated 
with  b  in  the  combination  (a  —  b)  +  b  is  equivalent  to  a, 
this  declaration,  or  the  equation 

(a  —  b)-\-b  =  a, 

is  a  definition  *  of  this  symbol. 

By  the  assumption  of  the  permanence  of  form  of  the 
numerical  equation  in  which  the  definition  of  subtraction 
resulted,  one  is  thus  put  immediately  in  possession  of  a 
symbolic  definition  of  subtraction  which  is  general. 

The  numerical  definition  is  subordinate  to  the  symbolic 
definition,  being  the  interpretation  oi  which  it  admits  when 
b  is  less  than  a. 

But  from  the  standpoint  of  the  symbolic  definition,  inter- 
pretability  —  the  question  whether  a  —  b  is  a  number  or 
not  —  is  irrelevant;  only  such  properties  may  be  attached 
to  a  —  b,  by  itself  considered,  as  flow  immediately  from  the 
generalized  equation 

{a  —  b)-\-b  =  a. 

In  like  manner  each  of  the  fundamental  laws  I-V,  VII, 
on  the  assumption  of  the  permanence  of  its  form  after  it 
has  ceased  to  be  interpretable  numerically,  becomes  a 
declaration  of  the  equivalence  of  certain  definite  combi- 
nations of  symbols,  and  the  formal  consequences  of  these 
laws  —  the  equations  -1-5  of  §  10  —  become  definitions 
of  addition,  subtraction,  multiplication,   and  their  mutual 

*  A  definition  in  terms  of  symbolic,  not  numerical  addition.  The 
sign  +  can;  of  course,  indicate  numerical  addition  only  when  "both 
the  symbols  which  it  connects  are  numbers. 


SUBTRACTION.  13 

relations  —  definitions  which  are  purely  symbolic,  it  may 
be,  but  unrestricted  in  their  application. 

Now  with  reference  to  the  legitimacy  of  such  definitions 
as  these  there  can  be  no  question.  They  are  consistent  with 
each  other,  and  of  course  consistent  with  the  numerical  defi- 
nitions, which  are  indeed  but  special  interpretations  of  them. 
If  used  consistently,  there  is  no  more  possibility  of  their  lead- 
ing to  false  results  than  there  is  of  the  more  tangible  numeri- 
cal definitions  leading  to  false  results.  The  laws  of  correct 
thinking  are  as  applicable  to  mere  symbols  as  to  numbers. 

What  the  value  of  these  symbolic  definitions  is,  to  what 
extent  they  add  to  the  power  to  draw  inferences  concerning 
numbers,  the  elementary  algebra  abundantly  illustrates. 

One  of  their  immediate  consequences  is  the  introduction 
into  algebra  of  two  new  symbols,  zero  and  the  negative, 
which  contribute  greatly  to  increase  the  simplicity,  compre- 
hensiveness, and  power  of  its  operations. 

13.  Zero.  When  b  is  set  equal  to  a  in  the  general 
equation 

(a  —  b)  +  b  =  a, 

it  takes  one  of  the  forms 

(a  —  a)  +  a  =  a, 
(b-b)  +  b  =  b. 
It  may  be  proved  that 

a  —  a  =  b  —  b. 
For  (a  —  a)  +  (a  +  b)  =  (a  —  a)  -f  a  +  b,      Law  II. 

=  a  +  by 
since  (a  —  a)  -{-  a  =  a. 

And  (b  -  b)  +  (a  +  b)  =  (b  -  b)  +6+a,  Laws  I,  II. 

=  b  +  a, 
since  (b  —  b)  -j-  b  =  b. 

Therefore  a  —  a  =  b  —  b.  Law  VII. 


14  NTJMBEB-SYSTEM  OF  ALGEBRA. 

a  —  a  is  therefore  altogether  independent  of  a  and  may 
properly  be  represented  by  a  symbol  unrelated  to  a.  The 
symbol  which  has  been  chosen  for  it  is  0,  called  zero. 

Addition  is  defined  for  this  symbol  by  the  equations 

1.  0  -f-  a  =  a7  definition  of  0. 
a  +  0  =  a.  Law  I. 

Subtraction  (partially) ,  by  the  equation 

2.  a  —  0  =  a. 

For         (a  -  0)  +  0  =  a.'  Def .  VI. 

Multiplication  (partially),  by  the  equations 

3.  ax0  =  0xa  =  0. 

For  a  X  0  =  a(b  —  6),  definition  of  0. 

=  ab  —  ab,  §  10,  5. 

=  0.  .  definition  of  0. 

14.  The  Negative.  When  b  is  greater  than  a,  equal  say 
to  a  +  d,  so  that  b  —  a  =  d,  then 

a  —  b  =  a  —  (a  +  d), 

=  a  —  a  —  d>  §  10,  1. 

=  0  —  d.  definition  of  0. 

For  0  —  d  the  briefer  symbol  —  d  has  been  substituted ; 
with  propriety,  certainly,  in  view  of  the  lack  of  significance 
of  0  in  relation  to  addition  and  subtraction.  The  equation 
0  —  d  =  —  d}  moreover,  supplies  the  missing  rule  of  sub- 
traction for  0.     (Compare  §  13,  2.) 

The  symbol  —  d  is  called  the  negative,  and  in  opposition 
to  it,  the  number  d  is  called  positive. 

Though  in  its  origin  a  sign  of  operation  (subtraction 
from  0),  the  sign  —  is  here  to  be  regarded  merely  as  part 
of  the  symbol  —  d. 

—  d  is  as  serviceable  a  substitute  for  a  —  b  when  a  <  b, 
as  is  a  single  numeral  symbol  when  a  >  b. 


SUBTRACTION.  15 

The  rules  for  reckoning  with  the  new  symbol — definitions 
of  its  addition,  subtraction,  multiplication  —  are  readily 
deduced  from  the  laws  I-V,  VII,  definition  VI,  and  the 
equations  1-5  of  §  10,  as  follows : 

1.  b  +  (-b)  =  -b  +  b  =  0: 

Tor  —  b  -f-  b  =  (0  —  b)  +  b,       definition  of  negative. 

=  0.  Def .  VI. 

—  b  may  therefore  be  defined  as  the  symbol  the  sum  of 
which  and  b  is  0. 

2.  a  +  (—b)  =  —  b  +  a  =  a  —  b. 

For  a  +  (  —  b)  =  a  +  (0  —  b),     definition  of  negative. 

=  an-  0-b,  §10,4. 

=  a-6.'  §13,1. 

3.  _  a +  (-&)  =  - (a +  &), 

For      —  a  +  (—  b)  =  0  —  a  —  6,  by  the  reasoning  in  §  14,  2. 

=  0-0  +  6),  §10,1. 

=  —  (a  -f-  6).        definition  of  negative. 

4.  a  —  (—  b)  =  a  +  b. 

For  a  —  (  —  b)  =  a  — :  (0  —  &),      definition  of  negative. 

=  a  -  0  +  b,  §  10,  2. 

=  a  +  6.  §  13,  2. 

5.  (-a)-(- &)  =  &-«. 

For      —  a  —  (—  6)  =  —  a  +  b,    by  the  reasoning  in  §  14,  4. 
=  b  -  a.  §  14,  2. 

Cor.  (-a)-(-a)  =  0. 

6.  a(  —  b)  =  (  —  b)a=  —  ab. 

For  0  =  a(6-6),  §13,3. 

=  ab  -f  a(  —  5).  Law  V. 

.-.  a(-  &)  =  -  a&.  '  §  14,  1 ;  Law  VII. 


16  NUMBEli-SYSTEM  OF  ALGEBRA. 

7.  (-a)xO  =  Ox(-a)=s=0. 

For  (-  a)  x  0  =  (  -  a) (b  -  6),  definition  of  0. 

=  (-a)6-(-a)6,  §  10,5. 

=  0.  §  14,  6,  and  5,  Cor. 

8.  (-a)(-6)  =  a&. 

For                        0  =  (— a)(6  — 6),  §14,7. 

=  (-  a) 6  -i-(- a)  (-  5),  Law  V. 

==_a&  +  (_a)(_&).  §14,8. 

...   (__  a)  (_  &)  =  05.                           §  14,  1 ;  Law  VII. 

By  this  method  one  is  led,  also,  to  definitions  of  equality 
and  greater  or  lesser  inequality  of  negatives.     Thus 

9.  —  a  >,  =  or  <  —  by . 

according  as       b  >,  =  or  <  a.* 

For  as  &>,=,<  a, 

_  a  +  a  +  b  >,  =,  <  -b  +  &  +  a,         §14,1;  §  13,  1. 
or  —  a  >,  =,  <  —  by  Law  VII  or  ^Il\ 

In  like  manner  —  a  <  0  <  b. 

15.  Recapitulation.  The  nature  of  the  argument  which 
has  been  developed  in  the  present  chapter  should  be  care- 
fully observed. 

From  the  definitions  of  the  positive  integer,  addition, 
and  subtraction,  the  associative  and  commutative  laws  and 
the  determinateness  of  subtraction  followed.  The  assump- 
tion of  the  permanence  of  the  result  a  —  b,  as  defined  by 
(a  —  &)  +  &==  a,  for  all  values  of  a  and  b,  led  to  definitions 

*  On  the  other  hand,  —  a  is  said  to  be  numerically  greater  than,  equal 
to,  or  less  than .—  6,  according  as  a  is  itself  greater  than,  equal  to,  or 
less  than  b. 


SUBTRACTION.  17 

of  the  two  symbols  0,  —  d,  zero  and  the  negative ;  and  from 
the  assumption  of  the  permanence  of  the  laws  I-V,  VII- 
were  derived  definitions  of  the  addition,  subtraction,  and 
multiplication  of  these  symbols, — the  assumptions  being 
just  sufficient  to  determine  the  meanings  of  these  operations 
unambiguously . 

In  the  case  of  numbers,  the  laws  I-V,  VII,  and  definition 
VI  were  deduced  from  the  characteristics  of  numbers  and 
the  definitions  of  their  operations ;  in  the  case  of  the  sym- 
bols 0,  —  d,  on  the  other  hand,  the  characteristics  of  these 
symbols  and  the  definitions  of  their  operations  were  deduced 
from  the  laws. 

With  the  acceptance  of  the  negative  the  character  of 
arithmetic  undergoes  a  radical  change.*  It  was  already  in 
a  sense  symbolic,  expressed  itself  in  equations  and  inequali- 
ties, and  investigated  the  results  of  certain  operations.  But ' 
its  symbols,  equations,  and  operations  were  all  interpretable' 
in  terms  of  the  reality  which,  gave  rise  to  it,  the  number  of. 
things  in  actually  existing  groups  of  things.  Its  connec- 
tion with  this  reality  was  as  immediate  as  that  of  the  ele- 
mentary geometry  with  actually  existing  space  relations* 

But  the  negative  severs  this  connection.  The  negative 
is  a  symbol  for  the  result  of  an  operation  which  cannot  be 
effected  with  actually  existing  groups' of  things,  which  is, 
therefore,  purely  symbolic.  And  not  only  do  the  fundamen- 
tal operations  and  the  symbols  on  which  they  are  performed 
lose  reality ;  the  equation,  the  fundamental  judgment  in  all 
mathematical  reasoning,  suffers  the  same  loss.  From  being 
a  declaration  that  two  groups  of  things  are  in  one-to-one 
correspondence,  it  becomes  a  mere  declaration  regarding 
two  combinations  of  symbols,  that  in  any  reckoning  one 
may  be  substituted  for  the  other. 

*  In  this  connection  see  §  25. 


18  NUMBER-SYSTEM  OF  ALGEBRA. 


III.    DIVISION  AND   THE  PKACTION. 

16.  Numerical  Division.  The  inverse  operation  to  multi- 
plication is  division. 

To  divide  a  by  &  is  to  find  a  number  which  multiplied  by 
b  produces  a.     The  result  is  called  the  quotient  of  a  by  b, 

and  is  written  —  •    By  definition 
P 

VIII.  f^)b  = 


Like  subtraction,  division  cannot  be  always  effected. 
Only  in  exceptional  cases  can  the  a-group  be  subdivided 
into  groups  each  containing  b  individuals. 

17.  Determinateness  of  Numerical  Division.  When  divis- 
ion can  be  effected  at  all,  it  can  lead  to  but  a  single  result ; 
it  is.  determinate. 

For  there  can  be  but  one  number  the  product  of  which 
by  b  is  a ;  in  other  words, 

IX  ( If  cb  =  db, 

X       c=d*  . 

For  b  groups  each  containing  c  individuals  cannot  be 
equal  to  b  groups  each  containing  d  individuals  unless 
c  =  d  (§4). 

This  is  a  theorem  of  fundamental  importance.  It  may  be 
called  the  law  of  determinateness  of  division.  It  declares 
that  if  a  product  and  one  of  its  factors  be  determined,  the 
remaining  factor  is  definitely  determined  also;  or  that  if 
one  of  the  factors  of  a  product  changes  while  the  other 
remains  unchanged,  the  product  changes.  It  alone  makes 
division  in  the  arithmetical  sense  possible.     The  fact  that 

*  The  case  b  =  0  is  excluded,  0  not  being  a  number  in  the  sense  in 
which  that  word  is  here  used. 


DIVISION  AND   THE  FBACTION.  19 

it  does  not  hold  for  the  symbol  0,  but  that  rather  a  product 
remains  unchanged  (being  always  0)  when  one  of  its  factors 
is  0,  however  the  other  factor  be  changed,  makes  division 
by  0  impossible,  rendering  unjustifiable  the  conclusions 
which  can  be  drawn  in  the  case  of  other  divisors. 

The  reasoning  which  proved  law  IX  proves  also  that 

(  As  cb  >  or  <  db, 

IX   .  -N 

(.  c  >  or  <  d. 

18.  Formal  Rules  of  Division.     The  fundamental  laws  of 
the  multiplication  of  numbers  are 

III.  ab  =  ba, 

IV.  a(bc)=ra$c, 

V.  a(b  +  c)  =ab  +  ac. 
Of  these,  the  definition 

VIII.  (r)b  =  a, 


the  theorem 

( If  ac  =  be, 
IX  ' 

\       a  =  b,  unless  c  =  0, 

and  the  corresponding  laws  of  addition  and  subtraction, 
the  rules  of  division  are  purely  formal  consequences,  dedu- 
cible  precisely  as  the  rules  of  subtraction  1-5  of  §  10  in  the 
preceding   chapter.     They  follow  without   regard  to   the 

meaning  of  the  symbols  a,  b,  c,  =,  +,  — ,  ab,  -•     Thus : 

b 

1. 

For 


and 

bd 


a    c  _ac 
b     d     bd 

?  .t  .bd  =  -b  - 
b     d             b 

d 

Laws  IV,  III. 

=  ac, 

Def.  VIII. 

—  •  bd  =  ac. 

Def.  VIII. 

20  NUMBER-SYSTEM  OF  ALGEBRA. 

The  theorem  follows  by  law  IX. 

2.  W  =  ad 

'c\      be 
d) 


For  ^s.c*  Def.  VIII. 

c\     d      b 

cl) 

and  —  •  -  =  -  •  - ,  §  18,  1 ;  Law  IV. 

be     d  .  b     cd 

_  a 

since  —  cd  =  dc  =  lx  cd.     Def.  VIII,  Law  IX. 

cd 

The  theorem  follows  by  law  IX. 

o  a  ,  c  _  adjkbc^ 

b      cl  bd 

Tor  f2  ±  5.^ M  =  -b  •  cZ±  -cZ .  b,  Laws III-V;  §  10, 5. 

\b      d)  b  d 

=  ad  ±  6c,  Def.  VIII. 

and  fad  ±  bc\d  =  ad±  be.  Def.  VIII. 

V     bd     J 

The  theorem  follows  by  law  IX. 

By  the  same  method  it  may  be  inferred  that 

b>'~'<d' 
as  ad  >,  =,  <  be.    Def.  VIII,  Laws  III,  IV,  IX,  IX'. 

19.  Limitations  of  Numerical  Division.  Symbolic  Division. 
The  Fraction.  General  as  is  the  form  of  the  preceding 
equations,  they  are  capable  of  numerical  interpretation  only 
when  -',  -  are  numbers,  a  case  of  comparatively  rare  occur- 
rence. The  narrow  limits  set  the  quotient  in  the  numer- 
ical definition  render  division  an  unimportant  operation  as 


DIVISION  AND   THE  FB ACTION.  21 

compared  with  addition,  multiplication,  or  the  generalized 
subtraction  discussed  in  the  preceding  chapter. 

But  the  way  which  led  to  an  unrestricted  subtraction  lies 
open  also  to  the  removal  of  this  restriction ;  and  the  reasons 
for  following  it  there  are  even  more  cogent  here. 

We  accept  as  the  quotient  of  a  divided  by  any  number  6, 

which  is  not  0,  the  symbol  -  defined  by  the  equation 

b 


regarding  this  equation  merely  as  a  declaration  of  the 
equivalence  of  the  symbols  (-)b  and  a,  of  the  right  to  sub- 
stitute one  for  the  other  in  any  reckoning. 

Whether  -  be  a  number  or  not  is  to  this  definition  irrele- 
b 

a 
vant*    When  a  mere  symbol,  t  is  called  a  fraction,  and  in 

opposition  to  this  a  number  is  called  an  integer. 

We  then  put  ourselves  in  immediate  possession  of  defi- 
nitions of  the  addition,  subtraction,  multiplication,  and 
division  of  this  symbol,  as  well  as  of  the  relations  of  equal- 
ity and  greater  and  lesser  inequality  —  definitions  which 
are  consistent  with  the  corresponding  numerical  definitions 
and  with  one  another  —  by  assuming  the  permanence  of 
form  of  the  equations  1,  2,  3  and  of  the  test  4  of  §  18  as 
symbolic  statements,  when  they  cease  to  be  interpretable  as 
numerical  statements. 

The  purely  symbolic  character  of  -  and  its  operations 

detracts  nothing  from  their  legitimacy,  and  they  establish 
division  on  a  footing  of  at  least  formal  equality  with  the 
other  three  fundamental  operations  of  arithmetic* 

*  The  doctrine  of  symbolic  division  admits  of  being  presented  in  the 
very  same  form  as  that  of  symbolic  subtraction. 

The  equations  of  Chapter  II  immediately  pass  over  into  theorems 


22  NUMBER-SYSTEM  OF  ALGEBRA. 

20.  Negative  Fractions.  Inasmuch  as  negatives  conform 
to  the  laws  and  definitions  I-IX,  the  equations  1,  2,  3  and 
the  test  4  of  §  18  are  valid  when  any  of  the  numbers  a,  b,  c,  d 
are  replaced  by  negatives.  In  particular,  it  follows  from  the 
definition  of  quotient  and  its  determinateness,  that 

a  __a m  —  a  _  _a .  —a_a 
^b~  P  ~V~  bl  —b~b' 
It  ought,  perhaps,  to  be  said  that  the  determinateness 
of  division  of  negatives  has  not  been  formally  demonstrated. 
The  theorem,  however,  that  if  (  ±  a)  (  ±  c)  =  (  ±  b)  (  ±  c), 
±  a  =  ±  b,  follows  for  every  selection  of  the  signs  ±  from 
the  one  selection  +,  +,  +,  +  by  §  14,  6,  8. 

21.  General  Test  of  the  Equality  or  Inequality  of  Fractions. 

Given  any  two  fractions   ±  -,   ±  — 

b        d 

±f>,=or<±^, 
b  d 

according  as  ±  ad  >,  =  or  <  ±  be. 

Laws  IX,  IX'.     Compare  §  4,  §  14,  9. 

respecting  division  when  the  signs  of  multiplication  and  division  are 
substituted  for  those  of  addition  and  subtraction  ;  so,  for  instance, 
a  —  (6  +  c)  =  a  —  6  —  c  =  a  —  c  —  b 

,    (i)  (?) 

gives  —  =  -l_i-  =  -l_u 

be         c  b 

In  particular,  to  (a  —  a)  +  a  =  a  corresponds  -  a  =  a.    Thus  a  purely 

symbolic  definition  may  be  given  1.  It  plays  the  same  role  in  multipli- 
cation as  0  in  addition.  Again,  it  has  the  same  exceptional  character 
in  involution  —  an  operation  related  to  multiplication  quite  as  multipli- 
cation to  addition  — as  0  in  multiplication;  for  1™  =  1",  whatever  the 
values  of  m  and  n. 

Similarly,  to  the  equation  (-  a)  +  a  =  0,  or  (0  -  a)  +  a  =  0,  corre- 
sponds  f-\  a  =  1,   which  answers  as  a  definition  of  the  unit  fraction 

-  ;  and  in  terms  of  these  unit  fractions  and  integers  all  other  fractions 

a 

may  be  expressed. 


DIVISION  AND   THE  FRACTION.  23 

22.  Indeterminateness  of  Division  by  Zero.  Division  by  0 
does  not  conform  to  the  law  of  determinateness ;  the  equa- 
tions 1,  2,  3  and  the  test  4  of  §  18  are,  therefore,  not  valid 
when  0  is  one  of  the  divisors. 

0  * 

The  symbols  -,  -,  of  which  some  use  is  made  in  mathe- 
matics, are  indeterminate.* 

1.  -  is  indeterminate.     For  -  is  completely  denned  by 

0  /0\ 

the  equation  (  -  j  0  ='  0 ;  but  since  x  x  0  =  0,  whatever  the 

value  of  x,  any  number  whatsoever  will  satisfy  this  equation. 

2.  -   is   indeterminate.       For,    by   definition,    j-JO  =  a. 

Were  -  determinate,  therefore,  —  since  then  ( - )  0  would, 

0                                   a  x  0  0  W 

by  §   18,    1,  be  equal  to ,  or  to   -,  —  the  number  a 

0        .  . 

would  be  equal  to  -,  or  indeterminate. 

Division  by  0  is  not  an  admissible  operation. 

23.  Determinateness  of  Symbolic  Division.  This  excep- 
tion to  the  determinateness  of  division  may  seem  to  raise 
an  objection  to  the  legitimacy  of  assuming —  as  is  done 
when  the  demonstrations  1-4.  of  §  18  are  made  to  apply  to 
symbolic  quotients  —  that  symbolic  division  is  determinate. 

It  must  be  observed,  however,  that  -,  -  are  indetermi- 
nate in  the  numerical  sense,  whereas  by  the  determinateness 
of  symbolic  division  is,  of  course,  not  meant  actual  numerical 
determinateness,  but  "  symbolic  determinateness,"  conform- 
ity to  law  IX,  taken  merely  as  a  symbolic  statement.  For, 
as  has  been  already  frequently  said,  from  the  present  stand- 
point the  fraction  -  is  a  mere  symbol,  altogether  without 
numerical  meaning  apart  from  the  equation  I  -  \b  =  a,  with 

*  In  this  connection  see  §  32. 


24  NUMBER-SYSTEM  OF  ALGEBRA. 

which,  therefore,  the  property  of  numerical  determinateness 
has  no  possible  connection.  The  same  is  true  of  the  prod- 
uct, sum  or  difference  of  two  fractions,  and  of  the  quotient 
of  one  fraction  by  another. 

As  for  symbolic  determinateness,  it  needs  no  justification 
when  assumed,  as  in  the  case  of  the  fraction  and  the 
demonstrations  1-4,  of  symbols  whose  definitions  do  not 
preclude  it.     The  inference,  for  instance,  that  because 


b  dj  \bd 

a  c  _ac 
b  d     bd 

which  depends  on  this  principle  of  symbolic  determinate- 
ness, is  of  precisely  the  same  character  as  the  inference 
that 

b  dj  b       d 

which  depends  on  the  associative  and  commutative  laws. 
Both  are  pure  assumptions  made  of  the  undefined  symbol 


a  c 
b  d 

with  that  of  the  product  of  two  numerical  quotients.* 


for  the  sake  of  securing  it  a  definition  identical  in  form 
b  d  5 


24.  The  Vanishing  of  a  Product.  It  has  already  been 
shown  (§  13,  3,  §  14,  7,  §  18,  1)  that  the  sufficient  condition 
for  the  vanishing  of  a  product  is  the  vanishing  of  one  of  its 
factors.  From  the  determinateness  of  division  it  follows 
that  this  is  also  the  necessary  condition,  that  is  to  say : 

If  a  product  vanish,  one  of  its  factors  must  vanish. 

Let  xy  —  0,  where  x,  y  may  represent  numbers  or  any  of 
the  symbols  we  have  been  considering. 


*  These  remarks,  mutatis  mutandis,  apply  with  equal  force  to  sub- 
traction. 


DIVISION  AND    THE  FRACTION.  25 

Since  xy  =  0, 

xy  +  xz  =  xz,  §  13,  1. 

or                                     x  (y  -\-z)  =  xz,  Law  V. 

whence,  if  x  be  not  0,          y  +  z  =  z>  Law  IX. 

or                                                y  =  0.  Law  VII. 

25.  The  System  of  Rational  Numbers.  Three  symbols, 
0    __  d  -    have  thus  been  found  which   can   be   reckoned 

with  by  the  same  rules  as  numbers,  and  in  terms  of  which 
it  is  possible  to  express  the  result  of  every  addition,  sub- 
traction, multiplication  or  division,  whether  performed  on 
numbers  or  on  these  symbols  themselves ;  therefore,  also, 
the  result  of  any  complex  operation  which  can  be  resolved 
into  a  finite  combination  of  these  four  operations. 

Inasmuch  as  these  symbols  play  the  same  r61e  as  numbers 
in  relation  to  the  fundamental  operations  of  arithmetic,  it 
is  natural  to  class  them  with  numbers.  The  word  "  number," 
originally  applicable  to  the  positive  integer  only,  has  come 
to  apply  to  zero,  the  negative  integer,  the  positive  and  nega- 
tive fraction  also,  this  entire  group  of  symbols  being  called 
the  system  of  rational  numbers*  This  involves,  of  course, 
a  radical  change  of  the  number  concept,  in  consequence  of 
which  numbers  become  merely  part  of  the  symbolic  equip- 
ment of  certain  operations,  admitting,  for  the  most  part,  of 
only  such  definitions  as  these  operations  lend  them. 

*  It  hardly  need  be  said  that  the  fraction,  zero,  and  the  negative 
actually  made  their  way  into  the  number- system  for  quite  a  different 
reason  from  this  ;  —  because  they  admitted  of  certain  "  real "  interpre- 
tations, the  fraction  in  measurements  of  lines,  the  negative  in  debit 
where  the  corresponding  positive  meant  credit  or  in  a  length  measured 
to  the  left  where  the  corresponding  positive  meant  a  length  measured 
to  the  right.  Such  interpretations,  or  correspondences  to  existing 
things  which  lie  entirely  outside  of  pure  arithmetic,  are  ignored  in  the 
present  discussion  as  being  irrelevant  to  a  pure  arithmetical  doctrine 
of  the  artificial  forms  of  number. 


26  NUMBER-SYSTEM  OF  ALGEBRA. 

In  accepting  these  symbols  as  its  numbers,  arithmetic 
ceases  to  be  occupied  exclusively  or  even  principally  with 
the  properties  of  numbers  in  the  strict  sense.  It  becomes 
an  algebra  whose  immediate  concern  is  with  certain  opera- 
tions defined,  as  addition  by  the  equations  a  +  b  —  b  +  a, 
g '  -f-  (p  +  c)  =  a  +  b  +  c,  formally  only,  without  reference  to 
the  meaning  of  the  symbols  operated  on.# 


IY.   THE   IEKATIOtfAL. 

26.   The  System  of  Rational  Numbers  Inadequate.     The 

system  of  rational  numbers,  while  it  sufhces  for  the  four  fun- 
damental operations  of  arithmetic  and  finite  combinations  of 
these  operations,  does  not  fully  meet  the  needs  of  algebra. 

The  great  central  problem  of  algebra  is  the  equation,  and 
that  only  is  an  adequate  number-system  for  algebra  which 
supplies  the  means  of  expressing  the  roots  of  all  possible 
equations.  The  system  of  rational  numbers,  however,  is 
equal  to  the  requirements  of  equations  of  the  first  degree 
only ;  it  contains  symbols  not  even  for  the  roots  of  such 
elementary  equations  of  higher  degrees  as  a?  ==  2,  x2  =  —  1. 

But  how  is  the  system  of  rational  numbers  to  be  enlarged 
into  an  algebraic  system  which  shall  be  adequate  and  at  the 
same  time  sufficiently  simple  ? 

The  roots  of  the  equation 

Xn  +  plXn~1  +J?2#*~*H hPn-lX  +Pn  ==  0 

*  The  word  "  algebra"  is  here  used  in  the  general  sense,  the  sense 
in  which  quaternions  and  the  Ausdehungslehre  (see  §§  127,  128)  are  alge- 
bras. Inasmuch  as  elementary  arithmetic,  as  actually  constituted, 
accepts  the  fraction,  there  is  no  essential  difference  between  it  and 
elementary  algebra  with  respect  to  the  kinds  of  number  with  which 
it  deals  ;  algebra  merely  goes  further  in  the  use  of  artificial  numbers. 
.The  elementary  algebra  differs  from  arithmetic  in  employing  literal 
symbols  for  numbers,  but  chiefly  in  making  the  equation  an  object  of 
investigation. 


THE  IRBATIONAL.  27 

are  not  the  results  of  single  elementary  operations,  as  are 
the  negative  of  subtraction  and  the  fraction  of  division  5  for 
though  the  roots  of  the  quadratic  are  results  of  "  evolution/' 
and  the  same  operation  often  enough  repeated  yields  the 
roots  of  the  cubic  and  biquadratic  also,  it  fails  to  yield 
the  roots  of  higher  equations.  A  system  built  up  as  the 
rational  system  was  built,  by  accepting  indiscriminately 
every  new  symbol  which  could  show  cause  for  recognition, 
would,  therefore,  fall  in  pieces  of  its  own  weight. 

The  most  general  characteristics  of  the  roots  must  be 
discovered  and  defined  and  embodied  in  symbols  —  by  a 
method  which  does  not  depend  on  processes  for  solving 
equations.  These  symbols,  of  course,  however  character- 
ized otherwise,  must  stand  in  consistent  relations  with  the 
system  of  rational  numbers  and  their  operations. 

An  investigation  shows  that  the  forms  of  number  neces- 
sary to  complete  the  algebraic  system  may  be  reduced  to 
two :  the  symbol  V—  1,  called  the  imaginary  (an  indicated 
root  of  the  equation  x2  -f-  1  =  0),  ancl  the  class  of  symbols 
called  irrational,  to  which  the  roots  of  the  equation  0^—2  =  0 
belong. 


27.  Numbers  Defined  by  Regular  Sequences.  X-The  Irra- 
tional. On  applying  to  2  the  ordinary  method  for  extracting 
the  square  root  of  a  number,  there  is  obtained  the  follow- 
ing sequence  of  numbers,  the  results  of  carrying  the  reck- 
oning out  to  0,  *1,  2,  3,  4,  ...  places  of  decimals,  viz. : 

1,  1.4,  1.41,  1.414,  1.4142,  ... 

These  numbers  are  rational ;  the  first  of  them  differs  from 
each  that  follows  it  by  less  than  1,  the  second  by  less  than' 

— ,  the  third  by  less  than ,  •••  the  nth  by  less  than 

10'  J  100'  J  lO'*1 

And    ■■,       is  a  fraction  which  may  be  made  less  than  any 
assignable  number  whatsoever  by  taking  n  great  enough. 

( 


28        •  NUMBER-SYSTEM  OF  ALGEBBA. 

This  sequence  may  be  regarded  as  a  definition  of  the 
square  root  of  2.  It  is  such  in  the  sense  that  a  term  may 
be  found  in  it  the  square  of  which,  as  well  as  of  each  fol- 
lowing term,  differs  from  2  by  less  than  any  assignable 
number. 


Any  sequence  of  rational  numbers 

a1?  a2,  a3,  •••  aM,  CL^+ly  "'  aM+^  "** 

in  ivhich,  as  in  the  above  sequence,  the  term  a^  may,  by  tak- 
ing fx  great  enough,  be  made  to  differ  numerically  from  each 
term  that  folloivs  it  by  less  than  any  assignable  number,  so 
that,  for  all  values  of  v,  the  difference,  aM+v  —  aM,  is  numerically 
less  than  8,  however  small  S  be  taken,  is  called  a  regular 
sequence. 

The  entire  class  of  operations  which  lead  to  regular 
sequences  may  be  called  regular  sequence-building.  Evolution 
is  only  one  of  many  operations  belonging  to  this  class. 

Any  regular  sequence  is  said  to  "define  a  number"  —  this 
"  number  "  being  merely  the  symbolic,  ideal,  result  of  the 
operation  which  led  to  the  sequence.  It  will  sometimes  be 
convenient  to  represent  numbers  thus  defined  by  the  single 
letters  a,  b,  c,  etc.,  which  have  heretofore  represented  posi- 
tive integers  only. 

After  some  particular  term  all  terms  of  the  sequence  a1? 
a2,  •••  may  be  the  same,  say  a.  The  number  defined  by  the 
sequence  is  then  a  itself.  A  place  is  thus  provided  for 
rational  numbers  in  the  general  scheme  of  numbers  which 
the  definition  contemplates. 

When  not  a  rational,  the  number  defined  by  a  regular 
sequence  is  called  irrational. 

The  regular  sequence  .3,  .33,  •••,  has  a  limiting  value,  viz., 

- ;  which  is  to  say  that  a  term  can  be  found  in  this  sequence 
which  itself,  as  well  as  each  term  which  follows  it,  differs 

from  -  by  less  than  any  assignable  number.    In  other  words, 
o 


THE  IBBATIONAL. 

the  difference  between  r  and  the  /xth  term  of  the  sequence 
o 

may  be  made  less  than  any  assignable  number  whatsoever 
by  taking  /x  great  enough.  It  will  be  shown  presently 
that  the  number  defined  by  any  regular  sequence,  a1?  a2,  ••• 
stands  in  this  same  relation  to  its  term  aM. 

28.  Zero,  Positive,  Negative.  In  any  regular  sequence 
a1}  a2,  "•  a  term  a^  may  always  be  found  which  itself,  as 
well  as  each  term  which  follows  it,  is  either 

(1)  numerically  less  than  any  assignable  number, 
or  (2)  greater  than  some  definite  positive  rational  number, 
or  (3)  less  than  some  definite  negative  rational  number. 

In  the  first  case  the  number  a,  which  the  sequence  de- 
fines, is  said  to  be  zero,  in  the  second  positive,  in  the  third 
negative. 

29.  The  Four  Fundamental  Operations.  Of  the  numbers 
defined  by  the  two  sequences  : 

a1?  a2,  a3,  •••  a^,  a^  +  i,  •••  a^  +  v,   •••, 

HD  H2>  Ps>  •••  Phi  fin  +  b  •*•  fin  +  vy  ••• 

(1)  The  sum  is  the  number  defined  by  the  sequence : 

al  +  A?  a2  +  /?2>    •  •  •   a/n  +  A>  <V  +  1  +  A  +  h    '"  aM  +  v  +  A  +  v,   .*. 

(2)  The  difference  is  the  number  defined  by  the  sequence : 

al         Ply   a2  P25    •  •  •    a|u  —  P^,   a^  +  1  —  PfJL  +  l>    •  •  •    O-fJL  +  v  —  P^  +  vi    •  •  • 

(3)  The  product  is  the  number  defined  by  the  sequence  : 

aiA  a2/fe   ...  a^A,  a^  +  i/3^  +  1,  ...  CLu  +  vP^  +  y,   ... 

(4)  T%e  quotient  is  the  number  defined  by  the  sequence : 

o-i     a2  a,x      aa  +  i  aa+r 

For  these  definitions  are  consistent  with  the  correspond- 
ing definitions  for  rational  numbers ;  they  reduce  to  these 
elementary    definitions,    in   fact,   whenever   the    sequences 


30  NUMBER-SYSTEM  OF  ALGEBBA. 

a1?  a2,  ...;  ft,  ft,  ...  either  reduce  to  the  forms  a,  a,  ...; 
ft  ft  ...  or  have  rational  limiting  values. 

They  conform  to  the  fundamental  laws  I-IX.  This  is 
immediately  obvious  with  respect  to  the  commutative,  asso- 
ciative, and  distributive  laws,  the  corresponding  terms  of 
the  two  sequences  c^ft,  a2ft,  ... ;  ftal7  fta2,  •••?  f°r  instance, 
being  identically  equal,  by  the  commutative  law  for  ra- 
tionals. 

But  again  division  as  just  defined  is  determinate.  For 
division  can  be  indeterminate  only  when  a  product  may 
vanish  without  either  factor  vanishing  (cf.  §  24)  ;  whereas 
aift?  o.2/32,  •"  can  define  0,  or  its  terms  after  the  nth  fall 
below  any  assignable  number  whatsoever,  only  when  the 
same  is  true  of  one  of  the  sequences  a1?  a2,  ... ;  ft,  ft,  ...# 

It  only  remains  to  prove,  therefore,  that  the  sequences 
(1),  (2),  (3),  (4)  are  qualified  to  define  numbers  (§  27). 

(1)  and  (2)  Since  the  sequences  a1?  a2,  ... ;  ft,  ft,  ...  are, 
by  hypothesis,  such  as  define  numbers,  corresponding  terms 
in  the  two,  aM,  ft,  may  be  found,  such  that 

a^  +  v  —  a^  is  numerically  <  8, 
and  ft*  +  v  —  ft  is  numerically  <  8, 

and,  therefore,  (a^  +  v  ±  ft  +  r)  —  (<v  ±  ft)  <  28, 
for  all  values  of  v,  and  that  however  small  8  may  be. 

Therefore  each  of  the  sequences  a:  -f-  ft,  a2  +  ft,  . . .  ; 
aj  —  ft,  a2  —  ft,  ...  is  regular. 

(3)  Let  a^  and  ft  be  chosen  as  before. 
Then  a^  +  v  ft  +  v  —  a^ft, 

since  it  is  identically  equal  to 

a^  +  v  (ft  +  v  —  ft*)  -f-  ft  (a/x  +  I/  —  a/x), 

*  It  is  worth  noticing  that  the  determinateness  of  division  is  here 
not  an  independent  assumption,  but  a  consequence  of  the  definition  of 
multiplication  and  the  determinateness  of  the   division  of  rationals. 
The  same  thing  is  true  of  the  other  fundamental  lavs  I-V,  VII. 
4 


THE  IBBATIONAL.  31 

is  numerically  less  than  a^  +  v  8  +  ft  8,  and  may,  therefore,  be 
made  less  than  any  assignable  number  by  taking  8  small 
enough ;  and  that  for  all  values  of  v. 

Therefore  the  sequence  ajft,  a2ft,  ...  is  regular. 

/  A\  •  afJi+  V   °-IX    tV  +  vRlA  P/A  +  VajU. 

Pn  +  v         ft  ft  +  vPn 

which  is  identically  equal  to 

Pfji  +  vjo-fji  +  v  —  <y)  —  0-fj.  +  v{piJL  +  v  —  ft)  < 

ft  +  *>ft 

By  choosing  a^  and  ft  as  before  the  numerator  of  this 
fraction,  and  therefore  the  fraction  itself,  may  be-  made  less 
than  any  assignable  number ;  and  that  for  all  values  of  v. 

Therefore  the  sequence  —,  — ,   •••  is  regular, 
ft  ft 

30.  Equality.     Greater  and  Lesser  Inequality.     Of  two 

numbers,  a  and  b,  defined  by  regular  sequences  aly  a2,  ... ;  ft,  ft, 
...,  the  first  is  greater  than,  equal  to  or  less  than  the  second, 
according  as  the  number  defined  by  ax  —  (3^  a2  —  ft,  ...  is 
greater  than,  equal  to  or  less  than  0. 

This  definition  is  to  be  justified  exactly  as  the  definitions 
of  the  fundamental  operations  on  numbers  defined  by  regu- 
lar sequences  were  justified  in  §  29. 

From  this  definition,  and  the  definition  of  0  in  §  28,  it 
immediately  follows  that 

Cor.  Two  numbers  ivhich  differ  by  less  than  any  assignable 
number  are  equal. 

31.  The  Number  Denned  by  a  Regular  Sequence  is  its 
Limiting  Value.  The  difference  between  a  number  a  and 
the  term  a^  of  the  sequence  by  which  it  is  defined  may  be 
made  less  than  any  assignable  number  by  taking  /x  great 
enough.. 


32  NUMBER-SYSTEM  OF  ALGEBRA. 

For  it  is  only  a  restatement  of  the  definition  of  a  regular 
sequence  a1?  a2,  ...  to  say  that  the  sequence 

which  defines  the  difference  a  —  a^  (§  29,  2),  is  one  whose 
terms  after  the  /xth  can  be  made  less  than  any  assignable 
number  by  choosing  /x  great  enough,  and  which,  therefore, 
becomes,  as  /x  is  indefinitely  increased,  a  sequence  which 
defines  0  (§  28). 

In  other  words,  the  limit  of  a  —  a^  as  /x  is  indefinitely 
increased  is  0,  or  a  =  limit  (<v).     Hence 

The  numb&QHefined  by  a  regular  sequence  is  the  limit  to 
which  the  fxth  term  of  this  sequence  approaches  as  /x  is  indefi- 
nitely increased* 

The  definitions  (1),  (2),  (3),  (4)  of  §  29  may,  therefore, 
be  stated  in  the  form : 

limit  (cty)  ±  limit  (^)  =  limit  (c^  ±  /^), 

limit  (<v)  •  limit  (&)=* limit  (a^), 

limit  Ok)-  =  limit  fe 

limit  (ft,)  ^ 


*  What  the  above  demonstration  proves  is  that  a  stands  in  the  same 
relation  to  a^  when  irrational  as  when  rational.  The  principle  of  perma- 
nence (cf.  §  12),  therefore,  justifies  one  in  regarding  a  as  the  ideal  limit  in 
the  former  case  since  it  is  the  actual  limit  in  the  latter  (§  27).    a,  when 

irrational,  is  limit  (a,*)  in  precisely  the  same  sense  that  -  is  the  quotient 

d 
of  c  by  c/,  when  c  is  a  positive  integer  not  containing  d.    It  follows  from 
the  demonstration  that  if  there  be  a  reality  corresponding  to  a,  as  in 
geometry  there  always  is  (see  §  40),  that  reality  will  be  the  actual 
limit  of  the  reality  of  the  same  kind  corresponding  to  a^. 

The  notion  of  irrational  limiting  values  was  not  immediately  avail- 
able because,  prior  to  §§  28,  29,  30,  the  meaning  of  difference  and  greater 
and  lesser  inequality  had  not  been  determined  for  numbers  defined  by 
sequences. 


THE  IEBATIONAL.  33 

For  limit  (a^)  the  more  complete  symbol    ^    (a^)  is  also 

used,  read  "the  limit  which  a^  approaches  as  fx  approaches 
infinity  "  ;  the  phrase  "  approaches  infinity  "  meaning  only, 
"  becomes  greater  than  any  assignable  number." 

32.   Division  by  Zero.     (1)   The  sequence  B?  f^  ...  cannot 

Pi    P<2 

define  a  number  when  the  number  defined  by  ftl9  f329  ...  is  0, 
unless  the  number  defined  by  a19  a2,  ...  be  also  0.  In  this 
case  it  may ;  -^  may  approach  a  definite  limit  as  //,  increases, 

however  small  a^  and  /^  become.  But  this  number  is  not  to 
be  regarded  as  the  mere  quotient  —     Its  value  is  not  at^U^ 

determined  by  the  fact  that  the  numbers  defined  by  al9  a2 . . . ; 
pl9  /32,  ...  are  0;  for  there  is  an  indefinite  number  of  dif- 
ferent sequences  which  define  0,  and  by  properly  choosing 
a1?  a2  . . . ;  f3l9  f32,  ...  from  among  them,  the  terms  of   the 

sequence  — ,  — ,  •  ••  may  be  made  to  take  any  value  what- 
soever.      ":    ^2 

(2)  The  sequence  — ,  — ,  ...  is  not  regular  when  /3l9  /329  ... 

ft     ft 

defines  0  and  al9  a2,  ...  defines  a  number  different  from  0. 
No  term  -^  can  be  found  which  differs  from  the  terms 

following  it  by  less  than  any  assignable  number ;  but 
rather,  by  taking  /x  great  enough,  -^  can  be  made  greater 

than  any  assignable  number  whatsoever. 

Though  not  regular  and  though  they  do  not  define 
numbers,  such  sequences  are  found  useful  in  the  higher 
mathematics.  They  may  be  said  to  define  infinity.  Their 
usefulness  is  due  to  their  determinate  form,  which  makes  it 
possible  to  bring  them  into  combination  with  other  sequences* 
of  like  character  or  even  with  regular  sequences. 

Thus   the   quotient   of    any   regular    sequence    yl9  y2   ... 


34  NUMBER-SYSTEM  OF  ALGEBRA. 

by  — ,  — ,   ...    is  a  regular   sequence   and  defines  0  ;    and 

the  quotient  of  —,  ,—,  ...  by  a  similar  sequence  2^  ^f,  ... 

fa    ft  $1    82 

may  also  be  regular  and  serve  —  if  ai}  fa,  y$  8f  (i  =  l,  2,  ...) 
be  properly  chosen  —  to  define  any  number  whatsoever. 

The  term  —  "approaches  infinity"  (i.e.  increases  with- 
out  limit)   as  /x  is   indefinitely  increased,  in  a  definite  or 

determinate  manner ;    so  that  the  infinity  which  — ,  — ,  ... 

Pi    fa 

defines  is  not  indeterminate  like  the  mere  symbol  -  of  §  22. 

But  here  again  it  is  to  be  said  that  this  determinateness 
is  not  due  to  the  mere  fact  that  fa,  p2  ...  defines  0,  which 

a 
is  all  that  the  unqualified  symbol  -  expresses.     For  there 

is  an  indefinite  number  of  different  sequences  which  like 

fa,  fa,  ...  define  0,  and  -  is  a  symbol  for  the  quotient  of  a 
by  any  one  of  them. 

33.  The  Number  System  denned  by  Regular  Sequences  of 
nationals,  a  Closed  System.    A  regular  sequence  of  irrationals 

a1?  a2,  ...  am,  amJrl,   ...  am+n,  ... 

(in  which  the  differences  am+n  —  am  may  be  made  numerically 
less  than  any  assignable  number  by  taking  .m  great  enough) 
defines  a  number,  but  never  a  number  which  may  not  also  be 
defined  by  a  sequence  of  rational  numbers. 

For  fa,  fa,  ...  being  any  sequence  of  rationals  which 
defines  0,  construct  a  sequence  of  rationals  aY,  a2,  ...  such 
that  a1—a1  is  numerically  less  than  fa  (§  30),  and  in  the  same 
sense  a2  —  a2</?2>  <^3  —  <*3<  fa,  etc.  Then  limit  (am— am)  =  0 
(§§  28,  31),  or  limit  (am)  =  limit  (am). 

This  theorem  justifies  the  use  of  regular  sequences  of 
irrationals  for  defining  numbers,  and  so  makes  possible  a 
simple  expression   of   the  results   of   some  very  complex 


THE  IMAGINARY.      COMPLEX  NUMBERS.  35 

operations.  Thus  am,  where  m  is  irrational,  is  a  number; 
the  number,  namely,  which  the  sequence  a**,  aaz,  ...  defines, 
when  ab  a2,  ...  is  any  sequence  of  rationals  defining  m. 

But  the  importance  of  the  theorem  in  the  present  discus- 
sion lies  in  its  declaration  that  the  number-system  defined 
by  regular  sequences  of  rationals  contains  all  numbers  which 
result  from  the  operations  of  regular  sequence-building  in 
general.  iTls  a  closed  system  with  respect  to  the  four 
fundamental  operations  and  this  new  operation,  exactly  as 
the  rational  numbers  constitute  a  closed  system  with  respect 
to  the  four  fundamental  operations  only  (cf.  §  25). 

The  number-system  defined  by  regular  sequences  of " 
(rationals  contains  every  number  which  lies  between  the 
extreme  limits  of  the  rational  number-system  (  —  cc  ,  +  ooj 
and  with  respect  to  whose  relation  to  each  and  every  num- 
ber of  that  system  it  can  be  said  that  it  is  either  greater 
than,  equal  to  or  less  than  fhat  number:  greater,  equal  or 
less  in  the  sense  in  which  one  rational  is  greater  than,  equal 
to,  or  less  than  another  (compare  §§  28,  30  and  §  21). 


Y.  THE   IMAGINAEY.     COMPLEX   NUMBEES. 

34.  The  Pure  Imaginary.  The  other  symbol  which  is 
needed  to  complete  the  number-system  of  algebra,  unlike 
the  irrational  but  like  the  negative  and  the  fraction,  admits 
of  definition  by  a  single  equation  of  a  very  simple  form,  viz., 

x2  +  1  =  0. 

It  is  the  symbol  whose  square  is  —  1,  the  symbol  V—  1, 
now  commonly  written  t\*  It  is  called  the  unit  of  imag- 
inaries. 

In  contradistinction  to  i  all  the  forms  of  number  hitherto 
considered  are  called  real.    These  names,  "real"  and  "imagi- 

*  Gauss  introduced  the  use  of  i  to  represent  V—  1. 


36  NUMBER-SYSTEM  OF  ALGEBRA. 

nary/''  are  unfortunate,  for  they  suggest  an  opposition  which 
does  not  exist.  Judged  by  the  only  standards  which  are 
admissible  in  a  pure  doctrine  of  numbers  i  is  imaginary  in 
the  same  sense  as  the  negative,  the  fraction,  and  the  irra- 
tional, but  in  no  other  sense  ;  all  are  alike  mere  symbols 
devised  for  the  sake  of  representing  the  results  of  opera- 
tions even  when  these  results  are  not  numbers  (positive 
integers),  i  got  the  name  imaginary  from  the  difficulty 
once  found  in  discovering  some  extra-arithmetical  reality 
to  correspond  to  it. 

As  the  only  property  attached  to  i  by  definition  is  that 
its  square  is  —  1,  nothing  stands  in  the  way  of  its  being 
" multiplied "  by  any  real  number  a;  the  product,  ia,  is 
called  &pure  imaginary. 

An  entire  new  system  of  numbers  is  thus  created,  coex- 
tensive with  the  system  of  real  numbers,  but  distinct  from 
it.  Except  0,  there  is  no  number  in  the  one  which  is  at 
the  same  time  contained  in  the  other. *  Numbers  in  either 
system  may  be  compared  with  each  other  by  the  definitions 
of  equality  and  greater  and  lesser  inequality  (§  30),  ia  being 

called   =  ib,  as  a  —  b :  but  a  number  in  one  system  cannot 

be  said  to  be  either  greater  than,  equal  to  or  less  than  a 
number  in  the  other  system. 

35.  Complex  Numbers.  The  sum  a  +  ib  is  called  a  com- 
plex number.  Its  terms  belong  to  two  distinct  systems,  of 
which  the  fundamental  units  are  1  and  i. 

The  general  complex  number  a  +  ib  is  defined  by  a  com- 
plex sequence 

«i  +  ipi,   a2  +  i(32,  . . .,  cv  +  ^73M,  . . ., 

where   a1?  a2,  ... ;  /?1?  /32,  ...    are  regular  sequences. 

*  Throughout  this  discussion  <x>  is  not  regarded  as  belonging  to  the 
number- system,  but  as  a  limit  of  the  system,  lying  without  it,  a  sym- 
bol for  something  greater  than  any  number  of  the  system. 


THE  IMAGINARY.      COMPLEX  NUMBERS.  37  ' 

Since  a  =  a  +  i'O  (§  36,  3,  Cor.)  and  ib  =  0  -f  t'6,  all  real 
numbers,  a,  and  pure  imaginaries,  ib,  are  contained  in  the 
system  of  complex  numbers  a  +  ib. 

a  +  ib  can  vanish  only  when  both  a  =  0  and  6  =  0. 

36.  The  Four  Fundamental  Operations  on  Complex  Num- 
bers. The  assumption  of  the  permanence  of  the  fundamen- 
tal laws  leads  immediately  to  the  following  definitions  of 
the  addition,  subtraction,  multiplication,  and  division  of 
complex  numbers. 

1.  (a  +  ib)  +  (a'  +  ib1)  =  a  +  a'  +  i(b  +  6') . 

For     (a  +  i'6)  +  (a'  +  #>')  =  a  +  i6  +  a'  +  ib',  Law  II. 

=  a  +  a'  +  i*6  +  ib',  Law  I. 

=  a+a'  +  i(6+6')%     Laws  II,  Y. 

2.  (a  +  ib)  -  (a'  +  ib')  =  a-  a[  +  i(b  -  6'). 

By  definition  of  subtraction  (VI)  and  §  36,  1. 

Cor.  The  necessary  as  well  as  the  sufficient  condition  for 
the  equality  of  tivo  complex  numbers  a  +  ib,  a'  +  ib'  is  that 
a  =  a'  and  b  =  b'. 

For  if    (a  +  ib)  -  (a'  +  ib1)  =a-a'  +  i(b  -6')  =  0,. 

a  -  a'  =  0,  &  -  6'  =  0  (§  35),  or  a  =  a',  6  =  6'.   \ 

3.  (a  +  ib)  (a'  +  i'6')  =*=  ana'  -  66'  +  i(ab'  +  ba'). 

For  (a  +  16)  (a'  +  iV)  =?*(a  +  16)  a'+  (a  -f  ib)'ib',  Law  V. 
==  aa'+ib  •  tx'  +  a  •  i'6'  +  i'6  •  ib',  Law  V. 
=  (aa'  —  66')  +  i(ab'+ba').  Laws  I-V. 

Cor.  ijf  either  factor  of  a  product  vanish,  the  product 
vanishes. 

For     ixO=  i(b  -  6)  =  ib  -  16  (§  10,  5),  =  0  (§  14,  1). 
Hence  (a  +  16)  0  =  C6  x  0  +  i6  X  0  =  a  x  0  +  i(&  X  0)  =  0. 

•     Laws  V,  IV,  §  28,  §  29,  3. 


38  NUMBER-SYSTEM  OF  ALGEBRA. 

a    a  +  ib  __  aa'  -f-  bb'      .ba'  —  abf 
'  a'  +  ib'~~  a'2+b'2       %  a'2  +  b'2 ' 
For  let  the  quotient  of  a  +  ib  by  a'  +  *&'  be  x  +  %• 
By  the  definition  of  division  (VIII), 

(x  -f  iy)  (a'-f-  #')  =  a  +  id. 
.-.  xa' —  yb' -\-i(xb' -\-ya')~  a -{-ib.  §36,3 

.-.  xa' —  yb'  =  a,    xb'  +  yaf  =  6.         §  36,  2,  Cor. 
Hence,  solving  for  x  and  ?/  between  these  two  equations, 
aa'  4-  bb'  ba'  —  ab' 


X'2  +  ^2  >         ^  a/2  +  b,2 

Therefore,  as  in  the  case  of  real  numbers,  division  is  a 
determinate  operation,  except  when  the  divisor  is  0;  it  is 
then  indeterminate.  For  x  and  y  are  determinate  (by  IX) 
unless  a'2  +  b'2  =  0,  that  is,  unless  a'  =  b'=  0,  or  a'-f-  ib'  =  0; 
for  a'  and  V  being  real,  a'2  and  b'2  are  both  positive,  and  one 
cannot  destroy  the  other. #     Hence, 'lay  the  reasoning  in  §  24, 

Cor.  If  a  product  of  tivo  complex  numbers  vanish,  one  of 
the  factors  must  vanish. 

37.   Numerical  Comparison  of  Complex  Numbers.      Two 

complex  numbers,  a  +  ib,  a'  +  ib',  do  not,  generally  speak- 
ing, admit  of  direct  comparison  with  each  other,  as  do  two 
real  numbers  or  two  pure  imaginaries  ;  for  a  may  be  greater 
than  a',  while  b  is  less  than  b'. 

They  are  compared  numerically,  however,  by  means  of 
their  moduli  Va2  -f  b2,   Va'2  +  b'2 ;    a  +  ib  being  said  to  be 

*  What  is  here  proven  is  that  in  the  system  of  complex  numbers 
formed  from  the  fundamental  units  1  and  i  there  is  one,  and  hut  one, 
number  which  is  the  quotient  of  a  +  ib  by  «'  -f  ib' ;  this  being  a  conse- 
quence of  the  determinateness  of  the  division  of  real  numbers  and 
the  peculiar  relation  (i2  =  —  1)  holding  between  the  fundamental 
units.  For  the  sake  of  the  permanence  of  IX  we  make  the  assumption, 
otherwise  irrelevant,  that  this  is  the  only  value  of  the  quotient  whether 
within  or  without  the  system  formed  from  the  units  1  and  u 


THE  IMAGINARY.      COMPLEX  NUMBERS.  39 

numerically  greater  than,  equal  to  or  less  than  a'  ■+-  ib! 
according  as  Va2  +  b2  is  greater  than,  equal  to  or  less  than 
Va'2  +  b'2.     Compare  §  47. 

38.  The  Complex  System  Adequate.  The  system  a  +  ib 
is  an  adequate  number-system  for  algebra.  For,  as  will  be 
shown  (Chapter  VII),  all  roots  of  algebraic  equations  are 
contained  in  this  system. 

But  more  than  this,  the  system  a  +  ib  is  a  closed  system 
with  respect  to  all  existing  mathematical  operations,  as  are 
the  rational  system  with  respect  to  all  finite  combinations 
of  the  four  fundamental  operations  and  the  real  system  with 
respect  to  these  operations  and  regular  sequence-building. 
For  the  results  of  the  four  fundamental  operations  on 
complex  numbers  are  complex  numbers  (§  36,  1,  2,  3,  4). 
Any  other  operation  may  be  resolved  into  either  a  finite 
combination  of  additions,  subtractions,  multiplications,  divis- 
ions or  such  combinations  indefinitely  repeated.  In  either 
case  the*  result,  if  determinate,  is  a  complex  number,  as  fol- 
lows from  the  definitions  1,  2,  3,  4  of  §  36,  and  the  nature 
of  the  real  number-system  as  developed  in  the  preceding 
chapter  (see  Chapter  VIII). 

The  most  important  class  of  these  higher  operations,  and 
the  class  to  which  the  rest  may  be  reduced,  consists  of 
those  operations  which  result  in  infinite  series  (Chapter 
VIII);  among  which  are  involution,  evolution,  and  the 
taking  of  logarithms  (Chapter  IX),  sometimes  included 
among  the  fundamental  operations  of  algebra. 

39.  Fundamental  Characteristics  of  the  Algebra  of  Num- 
ber. The  algebra  of  number  is  completely  characterized, 
formally  considered,  by  the  laws  and  definitions  I-IX 
and  the  fact  that  its  numbers  are  expressible  linearly  in 
terms  of  two  fundamental  units.*  It  is  a  linear,  asso- 
ciative, distributive,  commutative  algebra.     Moreover,  the 

*  That  is,  in  terms  of  the  first  powers  of  these  units. 


40  NUMBER-SYSTEM  OF  ALGEBRA. 

most  general  linear,  associative,  distributive,  commutative 
algebra,  whose  numbers  are  complex  numbers  of  the  form 
xfa  -h  x2e2  +  •  •  •  +  xnen,  built  from  n  fundamental  units  ely  e2, 
•  ••,  en,  is  reducible  to  the  algebra  of  the  complex  number 
a  -f-  ib.     For  Weierstrass  *  has  shown  that  any  two  complex 

numbers  a  and  b  of  the  form  x1e1  +  x2e2-h Yxr&m  whose 

sum,  difference,  product,  and  quotient  are  numbers  of  this 
same  form,  and  for  which  the  laws  and  definitions  I-IX  hold 
good,  may  by  suitable  transformations  be  resolved  into  com- 
ponents a1?  Og,  •  ••  a^ ;  &u  b%  •  •  •  br,  such  that 

a  =  a1-{-  a2-\ -f-  ar, 

6=^  +  ^+...  +  ^, 

a  ±  b  =  ax  ±  &J  -f  a2  ±  &2  -\ h  a,.  ±  br, 

ab  —  «!&!  +  «AH —  +  a  A, 

a  __  %      a2  ,         ,  ar 
&       &!       ft2  &r 

The  components  a,-,  5f  are  constructed  either  from  one  fun- 
damental unit  gt  or  from  two  fundamental  units  gi}  Avt 

For  components  of  the  first  kind  the  multiplication  for- 
mula is 

*  Zur  Theorie  der  aus  n  Haupteinheiten  gebildeten  complexen 
Grossen.     Gottinger  Nachrichten  Nr.  10,  1884. 

Weierstrass  finds  that  these  general  complex  numbers  differ  in  only- 
one  important  respect  from  the  complex  number  a  +  ib.  If  the  num- 
ber of  fundamental  units  be  greater  than  2,  there  always  exist  num- 
bers, different  from  0,  the  product  of  which  by  certain  other  numbers 
is  0.  Weierstrass  calls 'them  divisors  of  0.  The  number  of  exceptions 
to  the  determinateness  of  division  is  infinite  instead  of  one. 

t  These  units  are,  generally  speaking,  not  ev  e2, ...,  en,  but  linear 

combinations  of  them,  as  y^  +  y2e2  H J-  ynen,  k^  +  K2e2  -\ f-  Knen. 

Any  set  of  n  independent  linear  combinations  of  the  units  ev  e2, ...  en 
may  be  regarded  as  constituting  a  set  of  fundamental  units,  since  all 

numbers  of  the  form  a^  +  a2e2  ^ !-  anen  may  be  expressed  linearly 

in  terms  of  them. 


GBAPHICAL  REPRESENTATION  OF  NUMBERS.     41 

For  components  of  the  second  kind  the   multiplication 
formula  is 

(a9i  +  pkt)  (a'9i  +  p%)  =  (aa'-  PP^+W  +  &)K 

And  these  formulas  are  evidently  identical  with  the  mul- 
tiplication formulas 

(al)(j81)  =  (aj8)l, 
(al  +  pi)  (a'l  +  P'i)  =  (aa'-  pp')l  +  (a/3'  +  pj)% 

of  common  algebra. 


VI.    GEAPHICAL  EEPEESENTATION  OF  NUMBEES. 
THE  VAEIABLE. 

40.  Correspondence  between  the  Real  Number-System 
and  the  Points  of  a  Line.  Let  a  right  line  be  chosen,  and 
on  it  a  fixed  point,  to  be  called  the  null-point ;  also  a  fixed 
unit  for  the  measurement  of  lengths. 

Lengths  may  be  measured  on  this  line  either  from  left  to 
right  or  from  right  to  left,  and  equal  lengths  measured  in 
opposite  directions,  when  added,  annul  each  other ;  opposite 
algebraic  signs  may,  therefore,  be  properly  attached  to  them. 
Let  the  sign  -f-  be  attached  to  lengths  measured  to  the 
right,  the  sign  —  to  lengths  measured  to  the  left. 

The  entire  system  of  real  numbers  may  be  represented  by 
the  points  of  the  line,  by  taking  to  correspond  to  each  number 
that  point  whose  distance  from  the  null-point  is  represented 
by  the  number.  For,  as  we  proceed  to  demonstrate,  the 
distance  of  every  point  of  the  line  from  the  null-point, 
measured  in  terms  of  the  fixed  unit,  is  a  real  number; 
and  there  is  no  real  number  which  may  not  represent  such 
a  distance. 

1.  The  distance  of  any  point  on  the  line  from  the  null-point 
is  a  real  number. 

Let  any  point  on  the  line  be  taken,  and  suppose  the  seg- 
ment of  the  line  lying  between  this  point  and  the  null-point 


42  NUMBEB-SYSTEM  OF  ALGEBBA. 

to  contain  the  nnit  line  a  times,  with  a  remainder  dx,  this 
remainder  to  contain  the  tenth  part  of  the  unit  line  f$  times, 
with  a  remainder  d2,  d2  to  contain  the  hundredth  part  of  the 
unit  line  y  times,  with  a  remainder  d3,  etc. 

The  sequence  of  rational  numbers  thus  constructed,  viz., 
a,  a .  /?,  a .  /fy,  . . .  (adopting  the  decimal  notation)  is  regular ; 
for  the  difference  between  its  /xth  term  and  each  succeeding 

term  is  less  than  — — ,  a  fraction  which  may  be  made  less 

than  any  assignable  number  by  taking  p  great  enough ;  and, 
by  construction,  this  number  represents  the  distance  of  the 
point  under  consideration  from  the  null-point. 

By  the  convention  made  respecting  the  algebraic  signs  of 
lengths  this  number  will  be  positive  when  the  point  lies  to 
the  right  of  the  null-point,  negative  when  it  lies  to  the  left. 

2.  Corresponding  to  every  real  number  is  a  point  on  the 
line,  the  distance  of  which  from  the  nidi-point  is  represented 
by  the  number. 

This  is  immediately  evident  for  rational  numbers ;  a 
rational  length  may  be  actually  measured  off,  and  so  the 
point  be  actually  constructed. 

If  the  number  be  irrational,  let  a1?  a2?  ...  be  a  sequence  of 
rationals  defining  it.  There  is  a  point  on  the  line  which 
the  point  corresponding  to  the  term  a^  of  this  sequence 
approaches  as  limit  as  //.  is  indefinitely  increased,  and  whose 
distance  from  the  null-point  the  number  a,  defined  by  a1? 
a2,  ...,  represents. 

For  among  the  numbers  to  which  points  do  correspond 
(by  1),  one  can  be  found  which  is  equal  to  a.  For,  let  b 
(defined  by  /31}  f32  ...)  be  that  one  of  these  numbers  which 
differs  least  from  a.  If  this  difference  is  not  0,  in  the 
sequence  ^—(3^  a2—/327  ...  can  be  found  a  term  a^—fi^ 
which  itself,  as  well  as  each  term  aIJi  +  v—/3lji+v  following 
it,  is  either  greater  than  some  positive  rational  number 
8  or  less  than  some  negative  rational  number  —  S\     The 


GRAPHICAL  REPRESENTATION   OF  NUMBERS.     43 

number  b  */■  8  (or  b  -^  8')  differs  from  a  less  than  6  differs 
from  a;  and  a  point  corresponds  to  it,  namely,  the  point 
got  by  measuring  off  from  B  (which  by  hypothesis  cor- 
responds to  b)  the  rational  length  8  (or  —8').  Therefore, 
unless  b  is  equal  to  a,  among  the  numbers  to  which  points 
correspond  is  one  which  differs  less  from  a  than  b  does, 
which  is  contrary  to  hypothesis. 

41.   The  Real  Number-System  Continuous.     The  Variable. 

The  theorem  just  demonstrated  is  of  the  highest  impor- 
tance, for  it  establishes  the  right  to  represent  geometric 
magnitudes  by  numbers  and  to  discuss  geometric  relations 
algebraically. 

This  right  is  evidently  due  to  the  presence  of  the  irra- 
tional in  the  system  of  numbers.  The  geometric  magnitudes 
are  continuous ;  that  is  to  say,  the  boundary  separating  two 
contiguous  parts  of  a  geometric  magnitude  is  common  to 
both  these  parts.  For  instance,  the  point  C,  at  which  a 
given  line  AB  is  divided  into  the  segments  AC,  CB,  belongs 
to  both  of  these  segments.  It  is  altogether  different  with 
the  series  of  the  rational  numbers.  This  series  belongs  to 
the  class  of  discrete  magnitudes,  or  magnitudes  consecutive 
parts  of  which  have  distinct  boundaries ;  for,  between  any 
two  rational  numbers,  however  nearly  equal,  may  always  be 
inserted  an  irrational. 

The  entire  system  of  real  numbers,  however,  inasmuch  as 
it  contains  an  individual  number  to  correspond  to  every 
individual  point  in  the  continuous  series  of  points  forming 
a  right  line,  is  continuous. 

If  a  point  be  made  to  move  continuously  along  a  line,  its 
distance  from  any  fixed  point  on  the  line  will  run  through 
a  portion  of  this  continuous  number  series. 

Any  quantity  which  is  supposed  to  be  changing  is  called 
a  variable;  and  if,  like  the  distance  under  consideration,  its 
successive  values  form  a  continuous  series,  it  is  called  a 
continuous  variable. 


44 


NUMBER-SYSTEM  OF  ALGEBRA. 


B 


42.  Correspondence  between  the  Complex  Number-System 
and  the  Points  of  a  Plane.  The  entire  system  of  complex 
numbers  may  be  represented  by  the  points  of  a  plane,  as 
follows : 

In  the  plane  let  two  right  lines  X'OX  and  Y'OY  be 
drawn  intersecting  at  right  angles  at  the  point  0. 

Make  X'OX  the  "axis"  of  real  numbers,  using  its  points 
to  represent  real  numbers,  after  the  manner  described  in 

§  40,  and  make  Y'OY  the  axis 
of  pure  imaginaries,  represent- 
ing ib  by  the  point  of  OY 
whose  distance  from  0  is  b 
when  b  is  positive,  and  by  the 
corresponding  point  of  OY' 
when  b  is  negative. 
X  The  point  taken  to  represent 
the  complex  number  a+ib  is  P, 
constructed  by  drawing  through 
A  and  B,  the  points  which  rep- 
resent a  and  ib,  parallels  to 
Y'OY  and  X'OX,  respectively. 
The  correspondence  between  the  complex  numbers  and 
the  points  of  the  plane  is  a  one-to-one  correspondence.  To 
every  point  of  the  plane  there  is  a  complex  number  corre- 
sponding, and  but  one,  while  to  each  number  there  corre- 
sponds a  single  point  of  the  plane.*     * 

*  A  reality  has  thus  been  found  to  correspond  to  the  hitherto  unin- 
terpreted symbol  a  +  ib.  But  this  reality  has  no  connection  with  the 
reality  which  gave  rise  to  arithmetic,  the  number  of  things  in  a  group 
of  distinct  things,  and  does  not  at  all  lessen  the  purely  symbolic  char- 
acter of  a  -f  ib  when  regarded  from  the  standpoint  of  that  reality,  the 
standpoint  which  must  be  taken  in  a  purely  arithmetical  study  of 
the  origin  and  nature  of  the  number  concept. 

The  connection  between  the  numbers  a-f  ib  and  the  points  of  a 
plane  is  purely  artificial.  The  tangible  geometrical  pictures  of  the 
relations  among  complex  numbers  to  which  it  leads  are  nevertheless 
a  valuable  aid  in  the  study  of  these  relations. 


Fig.  1. 


GRAPHICAL  REPRESENTATION   OF  NUMBERS.    45 

It  follows,  by  the  reasoning  of  §  41,  that  the  system  of 
the  complex  numbers  is  a  continuous  system. 

If  the  point  P  be  made  to  move  about  in  its  plane,  the 
corresponding  number  runs  through  a  continuous  series  of 
complex  values,  and  is  called  a  complex  variable. 

43.  Modulus.  The  length  of  the  line  OP  (Fig.  1),  i.e. 
V«-  +  b2,  is  called  the  modulus  of  a  +  ib.  Let  it  be  repre- 
sented by  p. 

~ —  44.  Argument.  The  angle  XOP  made  by  OP  with  the 
positive  half  of  the  axis  of  real  numbers  is  called  the  angle 
of  a  +  ib,  or  its  argument.  Let  its  numerical  measure  be 
represented  by  0. 

The  angle  is  always  to  be  measured  "  counter-clockwise  " 
from  the  positive  half  of  the  axis  of  real  numbers  to  the 
modulus  line. 

45.   Sine.     The  ratio  of   PA,  the  perpendicular  from  P 

to  the  axis  of  real  numbers,  to   OP,  i.e.  -,  is  called  the 
sine  of  0,  written  sin  0.  P 

Sin  0  is  by  this  definition  positive  when  P  lies  above  the 
axis  of  real  numbers,  negative  when  P  lies  below  this  line. 

•  46.  Cosine.  The  ratio  of  PB,  the  perpendicular  from  P 
to  the  axis  of  imaginaries,  to  OP,  i.e.  -,  is  called  the  cosine 
of  0,  written  cos  0.  ? 

Cos  0  is  positive  or  negative  according  as  P  lies  to  the 
right  or  the  left  of  the  axis  of  imaginaries. 

47.  Theorem.  The  expression  of  a  +  ib  in  terms  of  its 
modulus  and  angle  is  p  (cos  0  +  i  sin  0) . 

For  by  §  46  -  =  cos  0,  .-.  a  =  p  cos  0 ; 

P 

and  by  §  45,  -  =  sin  0,   .-.  b  =  p  sin  0. 

P 
Therefore     a  +  ib  =  p  (cos  0  -f-  i  sin  0). 


46 


NVMBEB-SYSTEM  OF  ALGEBBA. 


The  factor  cos  0  +  i  sin  0  has  the  same  sort  of  geometrical 
meaning  as  the  algebraic  signs  +  and  — ,  which  are  indeed 
but  particular  cases  of  it :  it  indicates  the  direction  of  the 
point  which  represents  the  number  from  the  null-point. 

It  is  the  other  factor,  the  modulus  p,  the  distance  from 
the  null-point  of  the  point  which  corresponds  to  the  number, 
which  indicates  the  "  absolute  value "  of  the  number,  and 
may  represent  it  when  compared  numerically  with  other 
numbers  (§  37),  —  that  one  of  two  numbers  being  numeri- 
cally the  greater  whose  corresponding  point  is  the  more 
distant  from  the  null-point. 

48.  Problem  I.  Given  the  points  P  and  Pf,  representing 
a-j-ib  and  a'  +  ib'  respectively;  required  the  point  represent- 
ing a  +  af  -f-  i  (p  +  &')  • 

The  point  required  is  P",  the  intersection  of  the  parallel 
to  OP  through  P1  with  the  parallel  to  OPf  through  P. 

For  completing  the  construction  indicated  by  the  figure, 
we  have  OD1  =  PE  =  DD",  and  therefore  OD"  =  OD+  OD1 ; 
and  similarly  P"D"  =  PD  +  P'D\ 

Cor.  I.    To  get  the  point  corresponding  to  a  —  a'-t-i(b  —  b'), 

produce  OP'  to  P'"y  making 
OP'"=OP',  and  complete  the 
parallelogram  OP,   OP1". 

Cor.  II.  The  modulus  of  the 
sum  or  difference  of  two  complex 
numbers  is  less  than  {at  greatest 
equal  to)  the  sum  of  their  moduli. 

For  OP"  is  less  than  OP  + 
PP"   and,   therefore,  than    OP 
FlG<2>  +OP\  unless  0,  P,  P'  are  in 

the  same  •  straight  line,  when 
0P"=  0P+  OP1.  Similarly,  PP',  which  is  equal  to  the 
modulus  of  the  difference  of  the  numbers  represented  by 
P  and  P',  is  less  than,  at  greatest  equal  to,  OP  +  OP1. 


GBAPHICAL   REPRESENTATION   OF  NUMBERS.    47 

49.  Problem  II.  Given  P  and  P',  representing  a  +  ib 
and  a*  +  ib'  respectively ;  required  the  point  representing 
(a  +  ib)(a'  +ib'). 

Let  a -M&  =  p (cos  0  +  i'sin0),  §47 

and  a'  +ib'  =  p'(cos  &  +  *  sin  #')  5 

then  (a  +  ifyty'  +  iV) 

=  pp'(cos  0  +  £  sin  0)  (cos  0'  +  i  sin  0') 
=  pp'[_ (cos  0  cos  6'  —  sin  0  sin  0') 

+ 1 (sin  0  cos  0'  +  cos  0  sin  0')  ] . 
But  •  cos  (9  cos  (9'  —  sin  0sin0'  =  cos (0  +  0'),* 

and  sin  0  cos  0'  +  cos  0  sin  0'  =  sin (0  +  0')  * 

Therefore  (a+t&)  (a'  +  iV)=pp'[cos(0  +  0O+*'sin(0+0')] ; 
or,  The  modulus  of  the  product  of  two  complex  numbers  is 
the  product  of  their  moduli,  its  argument  the  sum  of  their 
arguments. 

The  required  construction  is,  therefore,  made  by  drawing 
through  0  a  line  making  an  angle  0  +  0'  with  OX,  and  lay- 
ing off  on  this  line  the  length  pp'. 

Cor.  I.  Similarly  the  product  of  n  numbers  having  moduli 
p,  p,  p",  •  ••  p(n)  respectively,  and  arguments  0,  0',  0",  ...  0(n), 
is  the  number 

pp'p"  ...  P<">[cos(0  +  0'  +  0"  +  ...  0<">) 

+  isin(0  +  0'  +  0"  +  ...0<n))]. 

In  particular,  therefore,  by  supposing  the  n  numbers  equal, 
we  may  infer  the  theorem 

[p  (cos  0  +  i  sin  0)  ]n  =  pn  (cos  nO  +  i  sin  nO) , 

which  is  known  as  Demoivre's  Theorem. 


*  For  the  demonstration  of  these,  the  so-called  addition  theorems  of 
trigonometry,  see  Wells'  Trigonometry,  §  65,  or  any  other  text-book 
of  trigonometry. 


48  NUMBER-SYSTEM  OF  ALGEBBA. 

Cor.  II.  From  the  definition  of  division  and  the  preceding 
demonstration  it  follows  that 

^|f  =  £[cos(.-^)  +  -in(^^)]; 

the  construction  for  the  point  representing    ^  is,  there- 

fore, obvious.  .  a'  + ib' 

50.  Circular  Measure  of  Angle.  Let  a  circle  of  unit  radius 
be  constructed  with  the  vertex  of  any  angle  for  centre. 
The  length  of  the  arc  of  this  circle  which  is  intercepted 
between  the  legs  of  the  angle  is  called  the  circular  measure 
of  the  angle. 

51.  Theorem.  Any  complex  number  may  be  expressed  in 
the  form  peie ;  ivhere  p  is  its  modulus  and  0  the  circular  meas- 
ure of  its  angle. 

It  has  already  been  proven  that  a  complex  number  may  be 
written  in  the  form  p(cos  0  +  ism  #),  where  p  and  0  have 
the  meanings  just  given  them.     The  theorem  will  be  demon- 
strated, therefore,  when  it  shall  have  been  shown  that 
eie  =  cos  0  -f  i  sin  0. 

If  n  be  any  positive  integer,  we  have,  by  §  36  and  the 
binomial  theorem, 

1   |  MY=1   |  nM  |  n(n-l)  (JO)' 

nj  n  2 !  ri1 

,  n(n-l)(n-2)  (W)3  , 


3! 
1 


=  1  +  0+      *  (*■*)« 


»8 


1- 

~2l 


i-m-2 


+± — ^ — ^w+. 

Let  n  be  indefinitely  increased ;  the  limit  of  the  right  side 
of  this  equation  will  be  the  same  as  that  of  the  left. 


GRAPHICAL  REPRESENTATION   OF  NUMBERS.    49 

But  the  limit  of  the  right  side  is 

(joy ,  (My , 


1  +  iO- 


21 


3! 


i.e.  e 


40  # 


20\n 


Therefore  e1'9  is  the  limit  of  f  1  H —  )  as  n  approaches  oo. 

\        nJ 

f         i0\ n 
To  construct  the  point  representing  ( 1  -\ —  )  : 

V        nJ 
On  the  axis  of  real  numbers 
.  lay  off  OA  =  1. 

Draw  AP  equal  to  0  and  par- 
allel to  OB,  and  divide  it  into  n 
equal  parts.  Let  AAY  be  one 
of  these  parts.     Then  AY  is  the 

point  1  H 

n 

Through   Ax   draw   AYA2 
right   angles   to    OAY  and  con- 
struct the  triangle  OA1A2  simi- 
lar to  OAAY. 

A2  is  then  the  point  [  1  + 


n  j 

For  AOA2  =  2AOA1; 

and  since       OA2 :  0A1 :  :  0AX :  OA,  and  OA  =  1, 
the  length     OA2  =  the  square  of  length  OAx.         (see  §  49) 

In  like  manner  construct  A3  to  represent  tl-\ —  j,  A±  for 

Let  7i  be  indefinitely  increased.  The  broken  line  AAYA2 
...  An  will  approach  as  limit  an  arc  of  length  0  of  the  circle 
of  radius  OA  and,  therefore,  its  extremity,  An,  will  approach 
as  limit  the  point  representing  cos  0  +  i  sin  0  (§  47). 


*  This  use  of  the  symbol  e**  will  be  fully  justified  in  §  73. 


50  NUMBER-SYSTEM   OF  ALGEBRA. 

f        i0\n 
Therefore  the  limit  of  (l-\ —  ]  as  n  is  indefinitely  in- 

V         nJ 
creased  is    cos  0  +  i  sin  0. 

Bnt  this  same  limit  has  already  been  proved  to  be  eie. 
Hence  eie  =  cos  0  -f-  i  sin  Q* 


VII.     THE  FUNDAMENTAL   THEOKEM   OP  ALGEBEA. 

52.  The  General  Theorem.     If 

w  =  a^n  +  a^1  +  a2zn~2  -{ 1-  an^z  +  an, 

where  n  is  a  positive  integer,  and  a0,  a1}  . . .,  an  any  numbers, 
real  or  complex,  independent  of  z,  to  each  value  of  z  corre- 
sponds a  single  value  of  w. 

We  proceed  to  demonstrate  that  conversely  to  each  value 
of  iv  corresponds  a  set  of  n  values  of  z,  i.e.  that  there  are 
n  numbers  which,  substituted  for  z  in  the  polynomial 
a&n  -f  ctiZ71'1  -f-  •••  +  an,  will  give  this  polynomial  any  value, 
to0,  which  may  be  assigned. 

It  will  be  sufficient  to  prove  that  there  are  n  values  of  z 

which  render  a0zn  +  a^z71"1  -\ +  an  equal  to  0,  inasmuch  as 

from  this  it  would  immediately  follow  that  the  polynomial 
takes  any  other  value,  iv0,  for  n  values  of  z ;  viz.,  for  the 
values  which  render  the  polynomial  of  the  same  degree, 
a0zn  +  a^1  -\ -f-  (an  —  w0),  equal  to  0. 

53.  Root  of  an  Equation.  A  value  of  z  for  which 
a02n  -f-  a^*-1  -f-  •  •  •  +  an  is  0  is  called  a  root  of  this  poly- 
nomial, or  more  commonly  a  root  of  the  algebraic  equation 

aoZn  +  c^z11-1  -\ h  an  =  0. 

*  This  demonstration  is  due  to  Dr.  F.  Franklin.  See  American 
Journal  of  Mathematics,  Vol.  VII,  p.  376. 


FUNDAMENTAL    THEOBEM  OF  ALGEBRA. 


51 


54.   Theorem.     Every  algebraic  equation  has  a  root. 
Given  the  equation  described  in  §  52, 

w  =  a$n  +  a$x~Y  +  a2zn~2  -f-  •  •  •  -f-  an_xz  +  an- 

We  are  to  demonstrate  that  in  the  system  of  complex 
numbers  there  is  a  value  which,  if  assigned  z,  will  render 
w  =  0 ;  or  for  which  the  point 
representing  w  in  the  plane 
of  complex  numbers  (the 
w-point  we  may  call  it)  will 
coincide  with  the  null-point. 

If  not,  let  P  be  a  point 
nearer  to  0  than  any  other 
with  which  the  w-point  can 
be  made  to  coincide  (or  at 
least  as  near  as  any  other). 

Through  P  draw  a  circle 
having  its  centre  in  the  null- 
point   0.     Then,  by  the  hy- 
pothesis made,  no  value  can  be  given  z  which  will  bring 
the  corresponding  w-point  within  this  circle. 

But  the  w-point  can  be  brought  within  this  circle. 

For,'z0  and  ivQ  being  the  values  of  z  and  w  which  corre- 
spond to  P,  change  z  by  adding  to  z0  a  small  increment  8, 
and  let  A  represent  the  consequent  change  in  w.  A  is 
defined  by  the  equation 

(w0  +  A)  =  a0(z0  +  8)"  +  Oi(%  +  S)'1"1 

.       +  a2  (z0  +  8)  n~2  +  •  •  •  +  an_Y  (z0  +  8)  +  an. 

On  applying  the  binomial  theorem  and  arranging  the 
terms  with  reference  to  powers  of  8,  the  right  member  of 
this  equation  becomes 

<WQn  +  a^o*"1  H h  an-i»o  +  an 

+  [na^*-1  +  (n  -  1)  b#f-*  +  •••  +  an_J  8 
+  terms  involving  82,  83,  etc. 


52  NUMBER-SYSTEM  OF  ALGEBRA. 

But  w0  =  a0zQn  +  a^-1  -\ 1-  an_^0  -f  an. 

.  • .  A  =  [nd^*"1  +  (n  -  1)  a^0n-2  +  •  •  •  +  an_2]  8 
-f-  terms  involving  S2,  S3,  etc. 
Let  p'  (cos  0'  -f  i  sin  0')  be  the  complex  number 

na^-1  -\-(n  —  l)  a^-2  -\ \-  an_^ 

expressed  in  terms  of  its  modulus  and  angle,  and 

p  (cos  0  +  isin0), 
the  corresponding  expression  for  S.     Then 

A  =  p'(cos  0'  +  t  sin  0')  X  p  (cos  0  -f  i  sin  0) 
-f-  terms  involving  p2,  p3,  etc. 
=  w>'[eos(0  +  0')  +  *sin  (0  +  0')] 

+  terms  involving  p2,  p3,  etc.      §  49. 

The  point  which  represents  pp' [cos (0  +  0')  -\-isui(6  +  0')] 
for  any  particular  value  of  p  can  be  made  to  describe  a 
circle  of  radius  pp'  about  the  null-point  by  causing  0  to 
increase  continuously  from  0  to  4  right  angles. 

In  the  same  circumstances  the  point  representing 

w0  +  pp'  [cos  (0  +  0')  +  i  sin  (0  +  0')  ] 

will  describe  an  equal  circle  about  the  point  P  and,  there- 
fore, come  within  the  circle  OP. 

But  by  taking  p  small  enough,  A  may  be  made  to  differ  as 
little  as  we  please  from  pp' [cos (0  +  0')  +  tsin(0  +  0')],#  and, 
therefore,  the  curve  traced  out  by  P'  (which  represents  w0-f  A, 
as  0  runs  through  its  cycle  of  values),  to  differ  as  little  as 
we  please  from  the  circle  of  centre  P  and  radius  pp'. 

Therefore  by  assigning  proper  values  to  p  and  0,  the  w- 
point  (P')  may  be  brought  within  the  circle  OP. 

*  In  the  series  Ap  +  Bp2  +  Cp3  +  etc. ,  the  ratio  of  all  the  terms  fol- 
lowing the  first  to  the  first,  i.e. 

Bp2  -f  Cp3  +  etc.    _     v  B  +  Cp  +  etc. . 

>  —  p  x  -  , 

Ap  A 

which  by  taking  p  small  enough  may  evidently  be  made  as  small  as 

we  please. 


FUNDAMENTAL    THEOREM  OF  ALGEBRA.  53 

The  to-point  nearest  the  null-point  must  therefore  be  the 
null-point  itself.* 

55.  Theorem.      If  a  be  a  root  of  a0zn  -f  a^-1  -\ h  an, 

this  polynomial  is  divisible  by  z  —  a. 

For  divide  a^zn  -f-  a^71"1  +  •  •  •  -f-  an  by  z  —  a,  continuing  the 
division  until  z  disappears  from  the  remainder,  and  call  this 
remainder  B,  the  quotient  Q,  and,  for  convenience,  the  poly- 
nomial f(z) . 

Then  we  have  immediately 

f(z)  =  (z-a)Q  +  B, 

holding  for  all  values  of  z. 

Let  z  take  the  value  a ;  then  f(z)  vanishes,  as  also  the 
product  (z  —  a)  Q. 

Therefore  when  z  =  a,  R  =  0,  and  being  independent  of  z 
it  is  hence  always  0. 

56.  The  Fundamental  Theorem.  The  number  of  the  roots 
of  the  polynomial  aQzn  -f-  a^z1^1  -f-  ...  -f-  an  is  n. 

For,  by  §  54,  it  has  at  least  one  root ;  call  this  a ;  then, 
by  §  55,  it  is  divisible  by  z  —  a,  the  degree  of  the  quotient 
being  n  —  1. 

Therefore  we  have 

o^+a^-M \-an=(z— a)(a0»n"1+&i»n-2H h&»~i)> 

Again,  by  §  54,  the  polynomial  a^71-1  +  bxzn-2  -\ h  bn_Y 

has  a  root ;  call  this  /?,  and  dividing  as  before,  we  have 

a^n + axzn~l + . . .  +a„=  (*r^»)  (z-0)  (a^n~2 + c^'3 + . . .  +  cn_2) . 


*  In  the  above  demonstration  it  is  assumed  that  the  coefficient  of  8, 
i.e.  na0z0n~l  -f  (n  —  l)^71"2  +  •••  +  «n-i,  is  not  0.  If  it  be  0,  it  is  only 
necessary  to  take  instead  of  P  some  other  point  on  the  circle  OP  ; 
na^-1  -f  etc.,  will  evidently  not  vanish  for  all  points  of  this  circle, 
since  the  number  of  its  roots  would  then  be  infinite  (see  §  56) . 


54  NUMBER-SYSTEM  OF  ALGEBRA. 

Since  the  degree  of  the  quotient  is  lowered  by  1  by  each 
repetition  of  this  process,  n  —  1  repetitions  reduce  it  to  the 
first  degree,  or  we  have 

ao^+a^-M \-an  =  a0(z-a)(z-P)(z-y)--'(z-v), 

a  product  of  n  factors,  each  of  the  first  degree. 

Now  a  product  vanishes  when  one  of  its  factors  vanishes 
(§  36,  3,  Cor.),  and  the  factor  z  —  a  vanishes  when  z  —  a, 
z  —  ft  when  z—ft,  •  •  •,  z  —  v  when  z  —  v.  Therefore  a0zn  +a12f~1 
+  •••  +  a>n  vanishes  for  the  n  values,  a,  ft,  y,  •••  v,  of  z. 

Furthermore,  a  product  cannot  vanish  unless  one  of  its 
factors  vanishes  (§  36,  4,  Cor.),  and  not  one  of  the  factors 
z  —  a,  z  —  ft,  •••,  z  —  .v,  vanishes  unless  z  equals  one  of  the 
numbers  a,  ft,  •  •  •  v. 

The  polynomial  has  therefore  n  and  but  n  roots. 

The  theorem  that  the  number  of  roots  of  an  algebraic 
equation  is  the  same  as  its  degree  is  called  the  fundamental 
theorem  of  algebra. 


VIII.     INFINITE  SEEIES. 

57.  Definition.  Any  operation  which  is  the  limit  of  ad- 
ditions indefinitely  repeated  produces  an  infinite  series.  We 
are  to  determine  the  conditions  which  an  infinite  series 
must  fulfil  to  represent  a  number. 

If  the  terms  of  a  series  are  real  numbers,  it  is  called  a 
real  series;  if  complex,  a  complex  seizes. 

I.    REAL  SERIES. 

58.  Sum.     Convergence.     Divergence.     An  infinite  series 

%  +  ^2  +  ^sH M»H 

represents  a  number  or  not,  according  as  the  sequence 

Sl)    S2)   S3)    '"    S/«?    Sm+1?    •"    Sm+n)    "*) 

where  sx  =  aly  s2—  «i  +  a2,  •  •  •,  st  =  a,  +  a2  -f  •  •  •  <%i, 

is  regular  or  not. 


INFINITE  SERIES.  55 

If  Sa  s2---,  be  a  regular  sequence,  the  number  which  it 
defines,  or    lim    (sn),  is  called  the  s?m  of  the  infinite  series 

a-i  +  a2  +  a3  + h«„H j 

and  the  series  is  said  to  be  convergent. 

If  sl7  s2,  be  not  a  regular  sequence,  sn  either  transcends 
any  finite  value  whatsoever,  as  n  is  indefinitely  increased, 
or  while  remaining  finite  becomes  altogether  indeterminate. 
The  infinite  series  then  has  nor  sum,  and  is  said  to  be  diver- 
gent. 

The  series  1  +  1  +  1  +  •••  and  1  —  1  +  1  —  1  +  •••  are  examples  of 
these  two  classes  of  divergent  series. 

A  divergent  series  cannot  represent  a  number. 

59.  General  Test  of  Convergence.  From  these  definitions 
and  §  27  it  immediately  follows  that : 

The  infinite  series  ax  +  a2  +  •••  +  am  -\ is  convergent  when 

m  may  be  so  taken  that  the  differences  sm+n  —  sm  are  numeri- 
cally less  than  any  assignable  number  8  for  cdl  values  of  n, 
where  sm  and  sm+n  are  the  sum  of  the  first  m  and  of  the  first 
m-\-n  terms  of  the  series  respectively. 

If  these  conditions  be  not  fulfilled,  the  series  is  divergent. 

The  limit  of  the  last  term  of  a  convergent  series  is  0 ;  for 
the  condition  of  convergence  requires  that  by  taking  m 
great  enough,  sm+1  —  sm,  i.e.  am+1,  may  be  found  less  than 
any  assignable  number.  But  it  is  not  to  be  assumed  con- 
versely that  a  series  is  convergent,  if  the  limit  of  its  last 
term  is  0 ;  other  conditions  have  also  to  be  fulfilled,  sm+n— sm 

must  be  less  than  8  for  all  values  of  n. 

*\                                 1     1 
Thus  the  limit  of  the  last  term  of  the  series  \-\ 1 1-  •••  is  0 ;  but, 

as  will  presently  be  shown,  this  is  a  divergent  series. 

60.  Absolute  Convergence.  It  is  important  to  distin- 
guish between  convergent  series  which  remain  convergent 
when  all  the  terms  are  given  the  same  algebraic  signs  and 


»s 


56  X UMBER-SYSTEM  OF  ALGEBRA. 

convergent  series  which  become  divergent  on  this  change  of 
signs.  Series  of  the  first  class  are  said  to  be  absolutely  con- 
vergent ;  those  of  the  second  class,  only  conditionally  con- 
vergent. 

Absolutely  convergent  series  have  the  character  of  ordinary 
sums;  i.e.  the  order  of  the  terms  may  be  changed  without 
altering  the  sum  of  the  series. 

For  consider  the  series       ux  +  a2  -f  a3  +  ••• 
supposed  to  be  absolutely  convergent  and  to  have  the  sum  S,  when 
the  terms  are  in  the  normal  order  of  the  indices. 

It  is  immediately  obvious  that  no  change  can  be  made  in  the  sum 
of  the  series  by  interchanging  terms  with  finite  indices  ;  for  n  may  be 
taken  greater  than  the  index  of  any  of  the  interchanged  terms.  Then 
Sn  has  not  been  affected  by  the  change,  since  it  is  a  finite  sum  and 
it  is  immaterial  in  what  order  the  terms  of  a  finite  sum  are  added ; 
and  as  for  the  rest  of  the  series,  no  change  has  been  made  in  the 
order  of  its  terms. 

But  ax  -f  a2  -f  a3  -f-  •  •  •  may  be  separated  into  a  number  of  infinite  series, 
as,  for  instance,  into  the  series  ax  +  a3  +  a5  +  .••  and  a2  +  cr4  +  a6  +  •  ••, 
and  these  series  summed  separately.  Let  it  be  separated  into  I  such 
series,  the  sums  of  which  —  they  must  all  be  absolutely  convergent,  as 
being  parts  of  an  absolutely  convergent  series  —  are  S^\  S&\  •••  SV\ 
respectively  ;  it  is  to  be  proven  that 

s=  m  +  #(2)  +  sw  +  -  +  s®. 

Let  8^\  S^\  •••  be  the  sums  of  the  first  m  terms  of  the  series  #(1\ 
S(2\  •  ••,  respectively. 

Then,  by  the  hypothesis  that  the  series  ax-\- a2-\-  •  ••  is  absolutely 
convergent,  m  may  be  taken  so  large  that  the  sum 

Sm+n  +  Sm+n  +  '"  +  ^+n 

shall  differ  from  S  by  less  than  any  assignable  number  5  for  all  values 
of  n  ;  therefore  the  limit  of  this  sum  is  S. 

But  again,  n  may  be  so  taken  that  S^     shall  differ  from  SW  by 

less  than  -,  S^2]    from  S(2)  by  less  than  -,  •  •  •  ;  and  therefore  the  sum 

Cn  +C  +  -  +  S%+n  from  S{1)  +  S{2)  +  -  +  S{1)  by  less  than  (^  ) I ; 

i.e.  by  less  than  8.    Hence  the  limit  of  this  sum  is  SW  +  #(2)  +  •••  +  SM. 
Therefore  S  and  #0)  +  SW  +  •••  +  S®  are  limits  of  the  same  finite 
sum  and  hence  equal. 


INFINITE  SERIES.  57 

61.  Conditional  Convergence.  On  the  other  hand,  the 
terms  of  a  conditionally  convergent  series  can  be  so  arranged 
that  the  sum  of  the  series  may  take  any  real  value  whatsoever. 

In  a  conditionally  convergent  series  the  positive  and  the  negative 
terms  each  constitute  a  divergent  series  having  0  for  the  limit  of  its 
last  term. 

If,  therefore,  C  be  any  positive  number,  and  Sn  be  constructed  by 
first  adding  positive  terms  (beginning  with  the  first)  until  their  sum  is 
greater  than  (7,  to  these  negative  terms  until  their  sum  is  again  less  than 
O,  then  positive  terms  till  the  sum  is  again  greater  than  (7,  and  so  on 
indefinitely  ;  the  limit  of  Sn,  as  n  is  indefinitely  increased,  is  C. 

62.  Special  Tests  of  Convergence.  1.  If  each  of  the  terms 
of  a  series  ax  +  a2  +  •  •  •  be  numerically  less  than  (at  greatest 
equal  to)  the  corresponding  term  of  an  absolutely  convergent 
series,  or  if  the  ratio  of  each  term  of  a1  -f-  a2  +  •  •  *  to  the  corre- 
sponding term  of  an  absolutely  convergent  series  never  exceed 
some  finite  number  C,  the  series  a1  +  a2  +  •  •  •  is  absolutely 
convergent. 

If  on  the  other  hand,  each  term  of  a1  +  a2  +  •  •  •  be  numeri- 
cally greater  than  (at  the  lowest  equal  to)  the  corresponding 
term  of  a  divergent  series,  or  if  the  ratio  of  each  term  of 

d\-\-  a2-\ to  the  corresponding  term  of  a  divergent  series  be 

never  numerically  less  than  some  finite  number  C!,  different 
from  0,  the  series  a1  -f-  a2  -f-  •••  is  divergent. 

2.    The  series  ax  —  a2  +  a3  —  a4  H ,  the  terms  of  which  are 

alternately  positive  and  negative,  is  convergent,  if  after  some 
term  a{  each  term  be  numerically  less  or,  at  least,  not  greater 
than  the  term  which  immediately  precedes  it,  and  the  limit  of 
an,  as  n  is  indefinitely  increased,  be  0. 

For  here 

Sm+n  —  Sm=:  (—  l)m[am+i  —  «m+2  +  ...   (—  l)"-^^]. 

The  expression  within  brackets  may  be  written  in  either  of  the  forms 

(am+i  —  am+2)  +  (am+3  —  am+4)  +  •••  (1) 

or  am+i  —  (am+2  —  am+3) .  (2) 


+  2A-lj' 


58  NUMBER-SYSTEM  OF  ALGEBRA. 

It  is  therefore  positive,  (1),  and  less  than  am+h  (2);  and  hence  by- 
taking  m  large  enough,  may  be  made  numerically  less  than  any  assign- 
able number  whatsoever. 

The  series  1 j f-  ...  is,  by  this  theorem,  con- 

,  2      3      4  '     J  ' 

vergent. 

3.  The  series  1-J \-~-\ h .■•••  is  divergent. 

Tor  the  first  2*  terms  after  the  first  may  be  written 

2+  \2^1  +  T+2j  +  U2  +  1  +  22  +  2  +  2*  +  3  +  2*  +  &  )  + 

+  f        1        +  —*—  +  ■■■ 

\2A-1  +  1      2A-1  +  2  2A-1  +  2A- 

where,  obviously,  each  of  the  expressions  within  parentheses  is  greater 

than  — 
2 
The  sum  of  the  first  2*  terms  after  the  first  is  therefore  greater  than 

-,  and  may  be  made  to  exceed  any  finite  quantity  whatsoever  by  tak- 

2 

ing  \  great  enough. 

This  series  is  commonly  called  the  harmonic  series. 

By  a  similar  method  of  |3roof  it  may  be  shown  that  the 

series  1  +  —  -f-  —  -\ is  convergent  if  p  >1. 

Here,    1  +  J-<A,   J.  +  i.  +  i  +  i^  i,  ;.«.  < [L\L., 

2p      3p       2p     4?      &>      6p       I*       4p  \2p  ) 

and  the  sum  of  the  series  is,  therefore,  less  than  that  of  the  decreasing 

2        /  2  \  2 
geometric  series  1  + J-  (  —  )  +•••. 

The  series  1 H — -f-  — -  -| is  divergent  if  p  <  1,  the  terms 

being  then  greater  than  the  corresponding  terms  of 

2      3 

4.  The  series  a1  +  a2  +  a3+".  is  absolutely  convergent  if 
after  some  term  of  finite  index,  at,  the  ratio  of  each  term  to 
that  which  immediately  precedes  it  be  numerically  less  than  1 
and,  as  the  index  of  the  term  is  indefinitely  increased,  approach 


INFINITE  SERIES.  59 

a  limit  which  is  less  than  1 ;  but  divergent,  if  this  ratio  and 
its  limit  be  greater  than  1. 

For —  to  consider  the  first  hypothesis  first  —  let  a  be  the  'greatest 
value  which  this  ratio  has  after  the  term  at.  By  the  hypothesis  a  is  a 
fraction. 

Then,  5i±l  <  a,  .\  ai+l  <  aia  ; 


-^  <  a,  .*.  ai+2  S  ai+ia  ^  a***1 


,2 


*i+l 


■   a*'+*     <  a,  /.  ai+k  <  ai+Ck-i)a  <  —  <  aia*. 

tf<+(ft_l) 

The  given  series  is  therefore  < 

Si  +  0<[a  +  a2  +  a3  +  —  ak  +  — ]• 
And  this  is  an  absolutely  convergent  series. 

For  a  +  a2  +  ~.  a*  +  -  =  fS^    (a  +  a2  +  ...  +  a-) 


_  limit    {a  —  an+i\ 
~n  =  cc    [    i_a    ) 

=  — - — ,  since  a  is  a  fraction. 


The  given  series  is  therefore  absolutely  convergent,  §  62,  1. 

The  same  course  of  reasoning  would  prove  that  the  series  is  diver- 
gent when  after  some  term  a*  the  ratio  of  each  term  to  that  which 
precedes  it  is  never  less  than  some  quantity,  a,  which  is  itself  greater 
than  1. 

When  the  limit  of  the  ratio  of  each  term  of  the  series  to 
the  term  immediately  preceding  it  is  1,  the  series  is  some- 
times convergent,  sometimes  divergent.  The  series  con- 
sidered in  §  62?  3  are  illustrations  of  this  statement. 

63.  Limits  of  Convergence.  An  important  application 
of  the  theorem  just  demonstrated  is  in  determining  what 
are  called  the  limits  of  convergence  of  infinite  series  of  the 
form 

Oq  -f-  axx  +  a2x?  -f-  a3xs  +  •  •  •, 


60 


NUMBEB-SYSTEM  OF  ALGEBBA. 


where  x  is  supposed  variable,  but  the  coefficients  a0,  a1?  etc., 
constants  as  in  the  preceding  discussion.  Such  a  series  will 
be  convergent  for  very  small  values  of  x,  if  the  coefficients 
be  all  finite,  as  will  be  supposed,  and  generally  divergent 
for  very  great  values  of  X)  and  by  the  limits  of  conver- 
gence of  the  series  are  meant  the  values  of  x  for  which  it 
ceases  to  be  convergent  and  becomes  divergent. 

By  the  preceding  theorem  the  series  will  be  convergent  if 
the  limit  of  the  ratio  of  any  term  to  that  which  precedes  it 
be  numerically  less  than  1 ;  i.e.  if 


limit 
n  =  oo 


M>n+lX 


or 


limit  fa 


'^x\  <1: 


that  is,  if  x  be  numerically  <  limit  ( — — ) ;  and  divergent,  if 
x  be  numerically  >  limit  I  — - 


1.    Thus  the  infinite  series 

am  +  mam  ^x  -\ * )-am~2xL  -f-  •  •  •, 

which  is  the  expansion,  by  the  binomial  theorem,  of 
(a-\-x)m  for  other  than  positive  integral  values  of  m,  is 
convergent  for  values  of  x  numerically  less  than  a,  diver- 
gent for  values  of  x  numerically  greater  than  a. 

For  in  this  case 

m(m  —  l)---(m  —  n  -f  1)~ 


limit   /  an  \  _  limi 


aX- 


in)  l 


m(m  —  ])-"(m  —  n) 


limit   /    w  n  +  1 ' 


imit        \        n  J 


limit 

'  n  =  oo 


-1  +  - 


INFINITE  SERIES.  61 


2.   Again,  the  expansion  of  ex,  i.e.  1  -f-  x  -\ 1-  ...;  is  con- 
vergent for  all  finite  values  of  x. 


For  here      limit  f-M=  limit    -il 


The  same  is  true  for  the  series  which  is  the  expansion 
of  ax. 

64.    Operations  on  Infinite   Series.     1.    The  sum  of  two 

convergent  series,  %  +  % -]-•••  and  bl-j-b2-\-  •  •-,  is  the  series 
(<2j  +&!)  +  (a2  +  b2)  +  ••• ;  and  their  difference  is  the  series 
(a1  —  b1)-\-(a2  —  b2)-{ . 

The  sum  of  the  series  ax  +  a2  +  •  ••  is  the  number  defined  by  s1$  s2, ..., 
and  the  sum  of  the  series  bl-\-b2-] —  is  the  number  defined  by  tv  t2, ..., 

where  s^  =  ax  +  a2  -\ \-at  and  ti=bl-\-b2-\ \-bL.    The  sum  of  the  two 

series  is  therefore  the  number  defined  by  sY  +  tv  s2  +  tv  •  «•,  §  29,  (1). 

But  if  Si  =  («!  +  &0  +  («2  +  b2)  H f-  (at  -f  bt),  we  have  St  =  st  +  U 

for  all  values  of  i.  This  is  immediately  obvious  for  finite  values  of  r, 
and  there  can  be  no  difference  between  Si  and  *<  +  tt  as  s  approaches  co, 
since  it  would  be  a  difference  having  0  for  its  limit. 

Therefore  the  number  defined  by  sx  +  tv  s2  -f  t2,  •  ••,  is  the  sum  of  the 
series  (ax  +  b{)  -f  (a2  +  62)  +  •  ••. 

2.   TAe  product  of  two  absolutely  convergent  series 

&i  +  a2H °m^  &i  +  b2  -\ 

is  the  series  a±bx  +  (ajj2  +  aj^i)  +  (aA  +  <^2  +  a3^i)  H 

+  O  A  +  «A-1  H h  a»-l&3  +  «  A)  H • 

Each  set  of  terms  within  parentheses  is  to  be  regarded  as  constitut- 
ing a  single  term  of  the  product ;  and  it  will  be  noticed  that  the  first  of 
them  consists  of  the  one  partial  product  in  which  the  sum  of  the  indices 
is  2,  the  second  of  all  in  which  the  sum  of  the  indices  is  3,  etc. 

By  §  29,  (3),  the  product  of  a,  +  a2  +  ...  by  \  +  b2  +  ...  is  J1™^  «*.)'• 
where  sn  and  tn  represent  the  sums  of  the  first  n  terms  of  ax  -f  a2  +  •••, 
L\  +  fyj  +  m"t  respectively. 

Suppose  first  that  the  terms  of  al  +  a2+">   and  bx  +  b2  H are 

all    positive.      Then    if    Sn    be    the    sum  of    the  first  n  terms  of 


62  NUMBER-SYSTEM  OF  ALGEBRA. 

a A  +  (a A  +  a2^i)  +  ••*»  and  m  represent  -  when  n  is  even  and  n  ~~ 
when  n  is  odd, 

evidently  sn^n  >  &  >  «**«. 

-o   ,  limit  ,    .  v       limit  /     .  x 

But  n  ±  ^   (Sntn)  =  n±ao   (Smtm). 

Therefore  XW;-^M>- 

If  the  terms  of  o^  +  a2  +  •••,  6j  +  62  +  •••  be  not  all  of  the  same  sign, 
call  the  sums  of  the  first  n  terms  of  the  series  got  by  making  all  the 
signs  plus,  snf  and  tn'  respectively  ;  also  Sn',  the  sum  of  the  first  n  terms 
of  the  series  which  is  their  product. 

Then  by  the  demonstration  just  given 

limit  (,S"„)=  limit  0V'»); 

n  =  go  v     nj      n  —  go  v    .    nn 
but  Sn  always  differs  from  sntn  by  ress  than  (at  greatest  by  as  much  as) 
S'n  from  s'j'n  ;  therefore,  as  before, 

limit  («„)= limit  (SA). 

n  =  go  v    nj      n  =  co  K  nnj 

3.    The  quotient  of  a2  +  a2  -\ by  bl-\-b2-\ does  not 

admit  of  simple  expression  in  terms  of  the  a/s  and  6/s.     It 

is  an   absolutely  convergent  series  when  al-\-a2-\ and 

bi  -f-  b2  -I-  •  •  •    are    absolutely    convergent    and    the   sum  of 
b\  +  b2  H is  not  0. 

II.    COMPLEX  SERIES. 

The  terms  sum,  convergent,  divergent,  have  the  same  mean- 
ings in  connection  with  complex  as  in  connection  with  real 
series. 

65.  General  Test  of  Convergence.  A  complex  series, 
<h  +  ^  +  •••>  **  convergent  wlien  the  modulus  of  sm+n  —  sTO  ma?/ 
6e  made  ?ess  £7ia?i  amy  assignable  number  8  5?/  taking  m  great 
enough,  and  that  for  all  values  of  n;  divergent,  when  this 
condition  is  not  satisfied.     See  §  48,  Cor.  II ;  §  59. 

66.  Of  Absolute  Convergence.     Let 

<h  +  a2  +  •••  be  a  complex  series, 
and  A1  +  A2-\ ,  the  series  of  the  moduli  of  its  terms. 


INFINITE  SERIES.  63 

If  the  series  A1  +  A2  -\ be  convergent,  the  series  ax  -\-a2-\ — 

ivill  be  convergent  also. 

For  the  modulus  of  the  sum  of  a  set  of  complex  numbers  is  less  than 
(at  greatest  equal  to)  the  sum  of  their  moduli  (§  48,  Cor.  II).  By 
hypothesis,  &m+n  —  Sm  is  less  than  any  assignable  number  8,  when 
Sm  =  A1  +  A2  +  •••  -f  Am,  etc.;  much  more  must  the  modulus  of 
sm-\-n  —  sm  be  less  than  8. 

The  converse  of  this  theorem  is  not  necessarily  trne ; 
and  a  convergent  series,  ax  -f-  a2  -f  •••,  is  said  to  be  absolutely 
or  only  conditionally  convergent,  according  as  the  series 
Ai  +  A2  +  *  *  *  is  convergent  or  divergent. 

67.  The  Region  of  Convergence  of  a  complex  series 

a0  -f-  axz  +  a2z2  +  •  •  •, 

that  is,  the  region  of  the  plane  of  complex  numbers  within 
which  the  point  representing  z  must  lie  if  the  series  is  to  be 
convergent,  is  a  circle  ivhose  centre  is  the  null-point  and  radius 
1  the  modulus  of  the  {numerically)  greatest  value  of  z  for  which 
the  series  converges. 

1.  For  every  point  within  this  circle  the  series  converges  absolutely. 

Let  Z  represent  the  numerically  greatest  value  of  z  for  which  the 
series  converges. 

Then  since  a0+  axZ '+  a2Z'2  ^ —  is  convergent,  anZn  approaches  0  as 
n  is  indefinitely  increased  ;  hence  a  number  M  can  be  found  whrich  is 
numerically  greater  than  any  term  of  the  series. 

Let  z  take  any  value  which  is  numerically  less  than  Z,  whose  cor- 
responding point,  therefore,  lies  within  the  circle  through  Z. 

The  terms  of  the  series  a0  +  axz  +  a2z2  -{ — ,  are  then  numerically 
less  than  the  corresponding  terms  of 

(for,  numerically,  M>atZ\  :.  m{~  ]>a^).    But  this  is  an  abso- 
lutely convergent  series  (§62,4). 

Hence  the  series  a0  +  aYz  -\ is  absolutely  convergent  for  all  values 

of  z  within  the  circle  through  Z  (§62,  1). 


64  NUMBER-SYSTEM  OF  ALGEBRA. 

2.  For  a  point  on  the  circumference  of  this  circle  the  series  may  be 
convergent  or  it  may  be  divergent.     Thus  the  circle  of  convergence  of 

the  series  1  +  |  -f  —  -] —  is  of  radius  unity,  and  the  series  is  convergent 
for  the  point  —  1,  divergent  for  -f  1. 

68.  Theorem.  The  following  is  a  theorem  on  which  many 
of  the  properties  of  functions  defined  by  series  depend. 

If  the  series  a0  +  axz  +  a2z2  H \-  anzn  -\ 

have  a  circle  of  convergence  greater  than  the  null-point  itself 
and  z  run  through  a  regular  sequence  of  values  zY,  z2,  ...  de- 
fining 0,  the  sum  of  all  terms  following  the  first,  viz., 

axz  +  a2z2  +  — h  anzn  +  •  •  • 

ivill  run  through  a  sequence  of  values  likeivise  regular  and 
defining  0  ;  or,  the  entire  series  may  be  made  to  differ  as  little 
as  one  chooses  from  its  first  term  a0. 

The  numbers  zv  z2,  •••  are,  of  course,  all  supposed  to  lie  within  the 
circle  of  convergence,  and  for  convenience,  to  be  real.  It  will  be  con- 
venient also  to  suppose  «i>*{>Zn  etc.;  i.e.  that  each  is  greater  than 
the  one  following  it. 

Since  a0  +  aLz  +  a2z2  +  •••  +  anzn  -{ 

converges  absolutely  for  z  =  zv  so  also  does 

aYz  +  a2z2  + M^n+-, 

and,  therefore,  ax  +  a2z  -\ —  +  anzn'-\-  ••♦. 

Hence  Ax  +  A2zY  -\ \-  Anzxn  H — 

(where  Ai  —  modulus  ai)  is  convergent,  and  a  number  M  can  be  found 
greater  than  its  sum. 

And  since  for  z  =  z2,  z3,  •  ••  the  individual  terms  of 

Al  +  Aji+  —  +  Af&+  — 

are  less  than  the  corresponding  terms  of  Ax  +  A2zx  +  •••  -f  Anz™  +  •••, 
this  series  and,  therefore,  modulus  (ax  -f  a2z  +  •••)  remain  always  less 
than  M  as  z  runs  through  the  sequence  of  values  z2,  z3,  •••. 

Hence  the  values  of  modulus  (axz  +  a2z2  +  •••)  which  correspond  to 
z  =  zv  z2  •••  constitute  a  regular  sequence  denning  0,  each  term  being 
numerically  less  than  the  corresponding  term  of  the  regular  sequence 
zxM,  z2M,  •  •  •  which  defines  0. 


INFINITE  SERIES.  65 

Cor.    The  same  argument  proves  that  if 

or  zm(am  +  <xm+1z +—)> 

be  the  sum  of  all  terms  of  the  series  from  the  (m  -f-  l)th  on, 

the  series  am  +  am+1z  -\ can  be  made  to  differ  as  little  as 

one  may  please  from  its  first  term  am. 

69.  Operations  on  Complex  Series.  The  definitions  of 
sum,  difference,  and  product  of  two  convergent  complex  series 
are  the  same  as  those  already  given  for  real  series,  viz. : 

1.  The   sum  of  two   convergent  series,    a^  a2-\ and 

°i  4-  b2  +  •  •  •,  is  the  series  (ax  +  bx)  +  (a2  +  b2)  +  •  •  • ;  their 
difference,  the  series  (c^  —  &x)  +  (a2  —  b2)  -f-  •  ••. 

For  if        s .  =  aY  +  a2  +  •  ••  +  a{  and  U  =  bx  +  b2  -f  •  ••  +  &<, 
modulus  [(5TO+n  ±  *m+n)  -  (sm  ±  tm)~\ 

<  modulus  (sm+n  -  sTO)  +  modulus  (fm+n  -  fm), 

and  may,  therefore,  be  made  less  than  any  assignable  number  by  tak- 
ing m  great  enough.  The  theorem  therefore  follows  by  the  reasoning 
of  §  64,  1. 

2.  The  product  of  two  absolutely  convergent  series, 

&i  +  &2  +  %  +  •  •  •  and  £\  +  b2  +  b3  H , 

is  the  series  a^  +  {a$2  +  a2&i)  +  (afis  +  a2£>2  +  asbi)  •  ••. 

For,  letting  5<  =  AY  +  A2  +  •••  +  A{  and  Ti  =  j8,  +  B2  +  •••  +  Bh 
where  Ait  Bt,  are  the  moduli  of  at-,  6;,  respectively,  and  representing 
by  <rn  the  sum  of  the  first  n  terms  of  the  series 

«A  +  («A  +  a26i)  +  ••• 
and  by  2«  the  sum  of  the  first  n  terms  of  the  series 

A1B1  +  (AlB2  +  A2B1)+.~, 
we  have  modulus  (sntn  —  <rn)  <  #n  Tn  —  2n. 

But  the  limit  of  the  right  member  of  this  inequality  (or  equation) 
is  0  (§  64,  2) ;  therefore 

limit  f    N       limit   ,    .  N 


66  NUMBER-SYSTEM  OF  ALGEBRA. 

IX.  THE  EXPONENTIAL  AND  LOGAKITHMIC  EUNCTIONS, 

UNDETERMINED    COEFFICIENTS.      INVOLUTION    AND 
EVOLUTION.      THE    BINOMIAL   THEOREM. 

70.  Function.  A  variable  w  is  said  to  be  a  function  of  a 
second  variable  z  for  the  area  A  of  the  z-plane  (§  42),  when 
to  the  z  belonging  to  every  point  of  A  there  corresponds  a 
determinate  value  or  set  of  values  of  iv. 

Thus  if  w  =  2z,  w  is  a  function  of  z.  For  when  «al,  w  =  2  ;  when 
2  =  2,  itf  =  4 ;  and  there  is  in  like  manner  a  determinate  value  of  w  for 
every  value  of  z.    In  this  case  A  is  coextensive  with  the  entire  z- plane. 

Similarly  w  is  a  function  of  z,  if 

w  =  a0+a1z  +  a2z2  +  •••  +  anzn  +  •••, 
so  long  as  this  infinite  series  is  convergent,  i.e.  for  the  portion  of  the 
2-plane  bounded  by  a  circle  having  the  null-point  for  centre,  and  for 
radius  the  modulus  of  the  greatest  value  of  z  for  which  the  series  con- 
verges. 

It  is  customary  to  use  for  w  when  a  function  of  z  the 
symbol /(z),  read  "function  z." 

71.  Functional  Equation  of  the  Exponential  Function. 

For  positive  integral  values  of  z  and  t,  az  •  a1  =  az+t.  The 
question  naturally  suggests  itself,  is  there  a  function  of  z 
which  will  satisfy  the  condition  expressed  by  this  equation, 
or  the  "functional  equation"  f(^)f{t)=f(z-\-t),  for  all 
values  of  z  and  t  ? 

We  proceed  to  the  investigation  of  this  question  and 
another  which  it  suggests,  not  only  because  they  lead  to  defi- 
nitions of  the  important  functions  az  and  logaz  for  complex 
values  of  a  and  z,  and  so  give  the  operations  of  involution, 
evolution,  and  the  taking  of  logarithms  the  perfectly  gen- 
eral character  already  secured  to  the  four  fundamental 
operations, — but  because  they  afford  simple  examples  of  a 
large  class  of  mathematical  investigations.* 

*  An  application  of  the  principle  of  permanence  (§  12)  is  involved 
in  the  use  of  functional  equations  to  define  functions.     The  equation 


UNDETEBMINED   COEFFICIENTS.  67 

72.  Undetermined  Coefficients.  In  investigations  of  this 
sort,  the  method  commonly  nsed  in  one  form  or  another  is 
that  of  undetermined  coefficients.  This  method  consists  in 
assuming  for  the  function  sought  an  expression  involving  a 
series  of  unknown  but  constant  quantities  —  coefficients,  — 
in  substituting  this  expression  in  the  equation  or  equations 
which  embody  the  conditions  which  the  function  must 
satisfy,  and  in  so  determining  these  unknown  constants 
'  that  these  equations  shall  be  identically  satisfied,  that  is  to 
say,  satisfied  for  all  values  of  the  variable  or  variables. 

The  method  is  based  on  the  following  theorem,  called 
"  the  theorem  of  undetermined  coefficients,"  viz. : 

If  the  series  A  +  Bz+Cz2-\ —  be  equal  to  the  series  A'  +  B'z 
4-  C'z2  -J-  •  ••  for  all  values  of  z  which  make  both  convergent, 
and  the  coefficients  be  independent  of  z,  the  coefficients  of  like 
powers  of  z  in  the  two  are  equal. 

For,  since 

A  +  Bz  +  Cz2  +  -'  =  A'  +  B'z+C'z2  +  -:, 

A  -  A'  +  (B  -  B')  z  +  (0  -  C)  z2  +  ...  =  0 

throughout  the  circle  of  convergence  common  to  the  two 
given  series  (§§  67,  69,  1). 

And  being  convergent  within  this  circle,  the  series 

A  -  A'+  (B  -  B')z  +  (C-  0)z2  +  ... 

azat  —  az+f,  for  instance,  only  becomes  a  functional  equation  when  its 
permanence  is  assumed  for  other  values  of  z  and  t  than  those  for  which  it 
has  been  actually  demonstrated. 

In  this  respect  the  methods  of  definition  of  the  negative  and  the 
fraction  on  the  one  hand,  and  the  functions  az,  loga  z,  on  the  other,  are 
identical ;  but,  while  the  equation  (a  —  6)  +  b  =  a  itself  served  as  defi- 
nition of  a  —  6,  there  being  no  simpler  symbols  in  terms  of  which 
a  —  b  could  be  expressed,  from  the  equation  aza{  —  az+t  a  series 
(§  73,  (4))  may  be  deduced  which  defines  az  in  terms  of  numbers  of 
the  system  a  +  ib. 


68  NUMBER-SYSTEM  OF  ALGEBRA. 

can  be  made  to  differ  as  little  as  we  please  from  its  first 
term,  A-A'(%  68). 

.-.  A- A'  =  0  (§  30,  Cot.),  ori  =  M 
Therefore 

(B-B')z+(C-C')z2  +  ..-  =  0 

throughout  the  common  circle  of  convergence,  and  hence 
(at  least,  for  values  of  z  different  from  0) 
B-B,  +  (C-C')z  +  -  =  0. 
Therefore  by  the  reasoning  which  proved  that 
A-A'=0,   B-B'  =  0,ovB  =  B'. 
In  like  manner  it  may  be  proved  that  C=  C,  D  =  D',  etc. 
Cor.    If    A  +  Bz  +  Ct+Dz2  +  Ezt  +  Ft2  +  - 

=  A'+  B'z  +  C't  +D'z2  +  E'zt  +  F't2  +  ... 

for  all  values  of  z  and  t  which  make  both  series  convergent,  and 
z  be  independent  of  t,  and  the  coefficients  independent  of  both  z 
and  t,  the  coefficients  of  like  powers  of  z  and  t  in  the  two  series 
are  equal. 

For,  arrange  both  series  with  reference  to  the  powers  of 

either  variable.      The  coefficients  of  like  powers  of   this 

variable  are  then  equal,  by  the  preceding  theorem.     These 

coefficients  are  series  in  the  other  variable,  and  by  applying 

the  theorem  to  each  equation  between  them  the  corollary  is 

demonstrated. 

i 

73.  The  Exponential  Function.     To  apply  this  method  to 

the  case  in  hand,  assume 

/(«)  =  A0  +  A,z  +  A2z2  -f- ...  +  Anzn  +  -., 

and  determine  whether  values  of  the  coefficients  A{  can  be 
found  capable  of  satisfying  the  "  functional  equation," 

for  all  values  of  z  and  t. 


THE  EXPONENTIAL  FUNCTION.  69 

On  substituting  in  this  equation,  we  have,  for  all  values 
of  z  and  t  for  which  the  series  converge, 

(A0+A,z+A2z2+..-Atizn+---)(A0+A1t+A.f+.--AHtn+--.) 

or,  expanding  and  arranging  the  terms  with  reference  to  the 
powers  of  z  and  t, 

AqA0  +  AYA$  +  AqAj  +  A2AqZ2  +  AYA$t  +  A0A2t2  H 

+  AVU-i#"H-  +An_kAkzn-Hk-{--+A0Antn 

+  ••• 

=  Aq  +  A1z  +  Ajt  +  A.jz2  +  2A2zt  +  A2t2  +  ... 

+  Anzn  +  Annzn~H  +  .  •  •  +  i«B"  V  +  •  •  •  +  AJT  +  •  •  •, 

where  ^(n  -  l)...(n  -  ^  + 1)^ 

*  &! 

Equating  the  coefficients  of  like  powers  of  z  and  £  in  the 
two  members  of  this  equation,  we  get 

An_kAk  equal  always  to  Annk. 

In  particular  A0A0  =  A0,  therefore  A0  =  l.     Also 

AYAX  =  2  A„     A2AY  =  3  ^t3, 

or,  multiplying  these  equations  together  member  by  member, 

A?  =  Ann\,  or  A  =  ^f 

A  part  of  the  equations  among  the  coefficients  are,  there- 
fore, sufficient  to  determine  the  values  of  all  of  them  in 
terms  of  the  one  coefficient  Ax.  But  these  values  will  satisfy 
the  remaining  equations  ;  for  substituting  them  in  the  gen- 
eral equation 

An_kAk  =  Annk, 

we  get  Ark    x ^1=4)1  y M(ft-l)-(M-fe  +  l) 

(n -*)!*!.  n!  Jc\  ' 

which  is  obviously  an  identical  equation. 


70    '  NUMBER-SYSTEM  OF  ALGEBRA. 

The  coefficient  AY  or,  more'  simply  written,  A,  remains 
undetermined. 

It  has  been  demonstrated,  therefore,  that  to  satisfy  equa- 
tion (1),  it  is  only  necessary  that/ (2)  be  the  sum  of  an 
infinite  series  of  the  form 

i+^+!^+!^+...?  (2) 

where  A  is  undetermined ;  a  series  which  has  a  sum,  i.e.  is 
convergent,  for  all  finite  values  of  z  and  A.     (§  63,  2,  §  66.) 

By  properly  determining  A,  f(z)  may  be  identified  with 
a%  for  any  particular  value  of  a. 

If  az  is  to  be  identically  equal  to  the  series  (2),  A  must 
have  such  a  value  that 

A2       As 

2! 31 

Let  e*=l+2+|!-  +  |!.  +  ...,  (3) 

where  e  =  1  +  1  +  i  +  ~  +  ■••  5  * 

Then  eA  =  l+ J. +  —  +  —  +  .... 

2  !      3  ! 

Therefore  a  =  6A  ; 
or,  calling  any  number  which  satisfies  the  equation 

ez  =  a 
the  logarithm  of  a  to  the  base  e  and  writing  it  loge  a, 
J.  =  loge  a. 

*  This  number  e,  the  base  of  the  Naperian  system  of  logarithms,  is 
a  "  transcendental "  irrational,  transcendental  in  the  sense  that  there 
is  no  algebraic  equation  with  integral  coefficients  of  which  it  can  be  a 
root  (see  Hermite,  Comptes  Eendus,  LXX VII).  ir  has  the  same  char- 
acter, as  Lindemann  proved  in  1882,  deducing  at  the  same  time  the 
first  actual  demonstration  of  the  impossibility  of  the  famous  old  prob- 
lem of  squaring  the  circle  by  aid  of  the  straight  edge  and  compasses 
only  (see  Mathematische  Annalen,  XX). 


THE  EXPONENTIAL  FUNCTION.  71 

Whence  finally, 

^  =  l+{logea)Z.+  ^f^  +  ^f^+..,         (4) 

a  definition  of  az,  valid  for  all  finite  complex  valnes  of 
a  and  z,  if  it  may  be  assumed  that  logea  is  a  number,  what- 
ever the  value  of  a. 

The  series  (3)  is  commonly  called  the  exponential  series, 
and  its  sum  ez  the  exponential  function.  It  is  much  more  use- 
ful than  the  more  general  series  (2),  or  (4),  because  of  its 
greater  simplicity ;  its  coefficients  do  not  involve  the  loga- 
rithm, a  function  not  yet  fully  justified  and,  as  will  be 
shown,  to  a  certain  extent  indeterminate.  Inasmuch,  how- 
ever, as  ez  is  a  particular  function  of  the  class  az,  az  is  some- 
times called  the  general  exponential  function,  and  series  (4) 
the  general  exponential  series. 

74.   The  Functions  Sine  and  Cosine.     It  was  shown  in 
§  51  that  when  0  is  a  real  number, 
ei9  =  cos  6  +  i  sin  0. 

But      <»^+*+m+m+m+... 

2!     41 

Therefore  (by  §  36,^2,  Cor.),  for  real  values  of  0 

eos  6  =  1  -  —  + ,  (5) 

2  !      4 !  v  ' 

fl3       ft5 
and  81110  =  0-^  +  ^-...,  (6) 

series  which  both  converge  for  all  finite  values  of  0. 
Though  cos  0  and  sin  0  only  admit  of  geometrical  interpre- 
tation when  0  is  real,  it  is  convenient  to  continue  to  use 
these  names  for  the  sums  of  the  series  (5)  and  (6)  when  0 
is  complex. 


72  NUMBER-SYSTEM  OF  ALGEBRA. 

75.  'Periodicity.  When  0  is  real,  evidently  neither  its 
sine  nor  its  cosine  will  be  changed  if  it  be  increased  or 
diminished  by  any  multiple  of  f our  right  angles,  or  2  it  ;  or, 
if  n  be  any  positive  integer, 

cos  (0  ±  2  mr)  =  cos  0,    sin  (0  ±  2  mr)  =  sin  0, 

and  hence  e{(6  £  2  »*)  =  ei0. 

The  functions  eie,  cos  0,  sin  0,  are  on  this  account  called 
periodic  functions,  with  the  modiolus  of  periodicity  2-k. 

76.  The  Logarithmic  Function.     If  z  =  ez  and  t  =  eT, 

zt  =  ezeT=ez+T,  §73 

or  loge  zt  =  loge  z  -+-  loge  t.  (7) 

The  question  again  is  whether  a  function  exists  capable 
of  satisfying  this  equation,  or,  more  generally,  the  "func- 
tional equation," 

f(zt)=m+/(t),  (8) 

for  complex  values  of  z  and  t. 
When  z  =  0,  (7)  becomes 

loge0  =  loge0-f  loge£, 

an  equation  which  cannot  hold  for  any  value  of  t  for  which 
loget  is  not  zero  unless  loge0  is  numerically  greater  than 
any  finite  number  whatever.     Therefore  loge0  is  infinite. 
On  the  other  hand,  when  z  =  1,  (7)  becomes 

loge£  =  logel+logc£, 

so  that  loge  1  is  zero. 

Instead,  therefore,  of  assuming  a  series  with  undetermined 
coefficients  for  f(z)  itself,  we  assume  one  for/(l-{-z),  setting 

f(l  +z)  =  Axz  +  A2z2  +  -  +  Anzn  +  -., 

and  inquire  whether  the  coefficients  A{  admit  of  values 
which  satisfy  the  functional  equation  (8)  for  complex 
values  of  z  and  t. 


THE  LOGARITHMIC  FUNCTION.  73 

Now         l  +  z  +  t  =  (l+z)(l+—t—\  identically. 
•••/[l+(z  +  0]=/(l+2)+/fl+:    ' 


1  +  zf 

or  A1(z+t)  +  A2(z  +  ty  +  -+An(z  +  t)n  +  - 

=  Axz  +  A2z2  +  -  +  Anzn  +  - 

+  A1(l+z)-H+Ai(l+z)-V+...+AJl+z)-nF+:: 

Equating  the  coefficients  of  the  first  power  of  t  (§  72) 
in  the  two  members  of  this  equation, 

A1+2Aiz  +  3Asz2  +  -  +  O  + 1)  An+1zn  +  •■■ 

=  A1(l-z+z2-z3  +  —  +  (-l)nz"+—)  ;  h&'jnuHly* 

whence,  equating  the  coefficients  of  like  powers  of  z, 

A1^A1}2Ai  =  -A1}:;  nA^i-iy^A,-, 

or  A=-4  ->  An=(-iy-^,.... 

A  n 

As  in  the  case  of  the  exponential  function,  a  part  of  the 
equations  among  the  coefficients  are  sufficient  to  determine 
them  all  in  terms  of  the  one  coefficient  AY.  But  as  in  that 
case  (by  assuming  the  truth  of  the  binomial  theorem  for 
negative  integral  values  of  the  exponent)  it  can  be  readily 
shown  that  these  values  will  satisfy  the  remaining  equa- 
tions also. 

The  series       *--+- +(-  l)n~1-  +  ••• 

2      3  n 

converges  for  all  values  of  z  whose  moduli  are  less  than  1 
(§  62,  f). 
For  such  values,  therefore,  the  function 

^-|2+-+(-l)n-1f+-)  (9) 

satisfies  the  functional  equation 

/[(!  +  *)(! +0]=/(l+z)+/(l  +  0- 


74  NUMBEB-SYSTEM  .OF  ALGEBBA. 

And  since       z  =  1  —  (1  —  z)  and  t  =  1  —  (1  —  t)f 

the  function        -  aIi-z^1^ '  +  ...  +  Q^¥+...\ 

satisfies  this  equation  when  written  in  the  simpler  form 

/(at)  =f(z)  +f(t),        . 
for  values  of  1  —  z  and  1  —  t  whose  moduli  are  both  less 
than  1. 

1.  Loge  b.  To  identify  the  general  function  /(l  +  2)  with 
the  particular  function  loge(l+3)  it  is  only  necessary  to 
give  the  undetermined  coefficient  A  the  value  1. 

For  since  loge(l+z)  belongs  to  the  class  of  functions 
which  satisfy  the  equation  (8), 

loge(l  +  z)  =  A(z-^+..X 

Therefore 

=i+^H+-)+2V2(z-l2+-Y+- 

But     el0se(i+*)  =  l+z. 
Hence 


z* 


l+,  =  l+^-|+...j+^-|  +  ...J+...; 

or?  equating  the  coefficients  of  the  first  power  of  z,  A  =  1. 

The  coefficients  of  the  higher  powers  of  z  in  the  right 
number  are  then  identically  0. 

It  has  thus  been  demonstrated  that  loge&  is  a  number 
(real  or  complex),  if  when  b  is  written  in  the  form  1+2, 
the  absolute  value  of  z  is  less  than  1.  To  prove  that  it  is  a 
number  for  other  than  such  values  of  b,  let  b  =  peie  (§  51), 
where  p,  as  being  the  modulus  of  b,  is  positive. 

Then  loge6  =  loge/o  +  i0, 

and  it  only  remains  to  prove  that  logep  is  a  number. 

Let  p  be  written  in  the  form  en  —  (en  —  p),  where  en  is  the 
first  integral  power  of  e  greater  than  p. 


THE  LOGARITHMIC  FUNCTION.  75 


Then  since  §  ■  en  -  (en  —  p)  =  en/l  —  e    n  p\ 

loge/o  =  logeen  +  log/l  -  e-^~J 

1 ^  j  is  a  number  since "  is  less  than  1. 


2.  Loga  b.  It  having  now  been  fully  demonstrated  that 
az  is  a  number  satisfying  the  equation  azaT  ==  az+T  for  all 
finite  values  of  a,  Z,  T\  let  az  =  z,  aT  =  t,  and  call  Z  the 
logarithm  of  z  to  the  base  a,  or  loga2,  and  in  like  manner 
T,  log.*. 

Then,  since      zt  =  azaT  =  az+r, 

•      loga(^)  =  logas+loga*, 
or  logaz  belongs,  like  logez,  to  the  class  of  functions  which 
satisfy  the  functional  equation  (8). 

Pursuing  the  method  followed  in  the  case  of  logeb,  it  will 

be  found  that  loga(l  -f  z)  is  equal  to  the  series  A(z  —  ^--\ ) 

when  A  = .     This  number   is   called  the  modulus  of 

logea 

the  system  of  logarithms  of  which  a  is  base. 

77.  Indeterminateness  of  loga.  Since  any  complex  num- 
ber a  may  be  thrown  into  the  form  peie, 

logea  =  loge/o  +  i<9.  (10) 

This,  however,  is  only  one  of  an  infinite  series  of  possible 
values  of  logea.     For,  since  eie  =  e*(*±2»»)  (§  75), 

logea  =  logePe*(0±2™O=  loge/o  +  i(0  ±2mr), 
where  n  may  be  any  positive  integer.     Loge<x  is,  therefore, 
to  a  certain  extent  indeterminate  ;  a  fact  which  must  be  care- 
fully regarded  in  using  and  studying  this  function.*     The 

*For  instance  \oge(zt)  is  not  equal  to  logez  +  loge*  for  arbitrarily 
chosen  values  of  these  logarithms,  but  to  logez  +  loge£  ±  i2mr,  where  n 
is  some  positive  integer. 


76  NUMBEB-SYSTEM  OF  ALGEBBA. 

value  given  it  in  (10),  for  which  n= 0,  is  called  its  principal 
value. 

When  a  is  a  positive  real  number,  0  =  0,  so  that  the  prin- 
cipal value  of  loge  a  is  real ;  on  the  other  hand,  when  a  is  a 
negative  real  number,  0  =  tt,  or  the  principal  value  of  loge  a 
is  the  logarithm  of  the  positive  number  corresponding  to  a, 

pluS   17T. 

78.  Permanence  of  the  Remaining  Laws  of  Exponents. 

Besides  the  law  aza*  =  az+t  which  led  to  its  definition,  the 
function  az  is  subject  to  the  laws  : 


1.                       (azy  =  azt. 

2.                      (ab)z  =  azbz* 

1.                       (azy  =  azt. 

For       az  =  (el0Sea)z  =  1  +  (logea)z  + 

(logea)V 

2! 

•••§73,(4) 

=  l+z\ogea  +  ± 

zlogea)2 
2! 

• 

—  gz1ogea^ 

§  73,  (3) 

.-.   (el°Z'a)z  =  ezlos*a,  and  loge 

az  =  z  log6a. 

From  these  results  it  follows  that 

(azy  =  elo^aZ)t 

_  etlogea* 

==  gteloge* 

=*  a*. 

2.                     (ab)z  =  azbz. 

For                  (ab)z  =  el0^ah^ 

—  gzlogeab 

g—  ezlogea  +  zlogeb      ■ 

§  76,  (7) 

_  ez\ogea  t  gzloge& 

§  73,  (1) 

=  a*  •  &*. 

*  —  —  a*-*   which  is  sometimes  included  among  the  fundamental 
a* 
laws  to  which  az  is  subject,  follows  immediately  from  azaf  =  az+f  by 

the  definition  of  division. 


INVOLUTION  AND  EVOLUTION.  77 

79.  Permanence  of  the  Remaining  Law  of  Logarithms. 

In  like  manner,  the  function  loga2  is  subject  not  only  to  the 

kW  log.(20  =  log.Z  +  lOg.«, 

but  also  to  the  law 

logaz?  =  tlogaz. 
For  z  =  a}°%«% 

and  hence  z*  =  (alogaZy 

=  atl0*"z.  §  78,  1 

80.  Evolution.  Consider  three  complex  numbers  £,  z,  Z, 
connected  by  the  equation     f z  =  z. 

This  equation  gives  rise  to  three  problems,  each  of  which 
is  the  inverse  of  the  other  two.  For  Z  and  £  may  be  given 
and  z  sought ;  or  £  and  z  may  be  given  and  Z  sought ;  or, 
finally,  z  and  Z  may  be  given  and  £  sought. 

The  exponential  function  is  the  general  solution  of  the 
first  problem  (involution),  and  the  logarithmic  function  of 
the  second. 

For  the  third  (evolution)  the  symbol  Vz  has  been  devised. 

This  symbol  does  not  represent  a  new  function ;  for  it  is 

defined  by  the  equation  (-^/z)z  =  z,  an  equation  which  is 

1 
satisfied  by  the  exponential  function  zz. 

Like  the  logarithmic  function,  Vz  is  indeterminate,  though 
not  always  to  the  same  extent.  When  Z  is  a  positive 
integer,  £z  =  z  is  an  algebraic  equation,  and  by  §  56  has  Z 
roots  for  any  one  of  which  Vz  is,  by  definition,  a  symbol. 
From  the  mere  fact  that  z  =  t,  therefore,  it  cannot  be  in- 
ferred that  ~\/z  =  -\/t,  but  only  that  one  of  the  values  of 

-yjz  is  equal  to  one  of  the  values  of  V£     The  same  remark, 

1     1 
of  course,  applies  to  the  equivalent  symbols  zz,  t*. 

81.  Permanence  of  the  Binomial  Theorem.     By  aid  of  the 

results  just  obtained,  it  may  readily  be  demonstrated  that 
the  binomial  theorem  is  valid  for  general  complex  as  well 
as  for  rational  values  of  the  exponent. 


78  JSfUMBEB-STSTEM  OF  ALGEBRA. 

For  b  being  any  complex  number  whatsoever,  and  the 
absolute  value  of  z  being  supposed  less  than  1, 

(l+zy=eblos*{1+z) 

=  e  v     2     / 

=  l-f-6^+terms  involving  higher  powers  of  z. 

Therefore  let 

-  (l  +  z)h  =  1  +  bz  +  Astf  +  ---  +  AjT  +  -•*.        (11) 

Since,  then,  (a  +  z)h  =  cfifl  +  -Y  §  78,  2 

if  -  be  substituted  for  2;  in  (11),  and  the  equation  be  multi- 
a 

plied  throughout  by  a6, 

(a  +  z)b  =  a*  +  &a*-*s  +  A2ab~2z2  H h  ^""2"  +  •  *  •  •    (12) 

Starting  with  the  identity 

(i  +  z  +  ty=(i  +  z  +  ty, 

developing  (l+z  +  t)b  by  (11)  and  (l+z  +  t)h  by  (12), 
equating  the  coefficients  of  the  first  power  of  t  in  these 
developments,  multiplying  the  resultant  equation  by  1  +  zy 
and  equating  the  coefficients  of  like  powers  of  z  in  this 
product,  equations  are  obtained  from  which  values  may  be 
derived  for  the  coefficients  At  identical  in  form  with  those 
occurring  in  the  development  for  (1  +  z)h  when  &  is  a  posi- 
tive integer. 

It  may  also  be  shown  that  these  values  of  the  coefficients 
satisfy  the  equations  which  result  from  equating  the  coeffi- 
cients of  higher  powers  of  t. 


II. 

HISTOEIOAL. 


83 


I.    PEIMITIYE  NUMEKALS. 

82.  Gesture  Symbols.  There  is  little  doubt  that  primitive 
counting  was  done  on  the  fingers,  that  the  earliest  numeral 
symbols  were  groups  of  the  fingers  formed  by  associating 
a  single  finger  with  each  individual  thing  in  the  group  of 
things  whose  number  it  was  desired  to  represent. 

Of  course  the  most  immediate  method  of  representing 
the  number  of  things  in  a  group  —  and  doubtless  the  method 
first  used  —  is  by  the  presentation  of  the  things  themselves 
or  the  recital  of  their  names.  But  to  present  the  things 
themselves  or  to  recite  their  names  is  not  in  a  proper 
sense  to  count  them ;  for  either  the  things  or  their  names 
represent  all  the  properties  of  the  group  and  not  simply  the 
number  of  things  in  it.  Counting  was  first  done  when  a 
group  was  used  to  represent  the  number  of  things  in  some 
other  group ;  of  that  group  it  would  represent  the  number 
only  and,  therefore,  be  a  true  numeral  symbol,  which  it  is 
the  sole  object  of  counting  to  reach. 

Counting  ignores  all  the  properties  of  a  group  except  the 
distinctness  or  separateness  of  the  things  in  it  and  pre- 
supposes whatever  intelligence  is  required  consciously  or 
unconsciously  to  abstract  this  from  its  remaining  properties. 
On  this  account,  that  group  serves  best  to  represent  num- 
bers, in  which  the  individual  differences  of  the  members 
are  least  obtrusive.  The  naturalness  of  finger-counting, 
therefore,  lies  not  only  in  the  accessibility  of  the  fingers, 
in  their  being  always  present  to  the  counter,  but  in  this : 
that  the  fingers  are  so  similar  in  form  and  function  that  it 
is  almost  easier  to  ignore  than  to  take  account  of  their 
differences. 

But  there  is  other  evidence  than  its  intrinsic  probability 

81 


NUMBER-SYSTEM  OF  ALGEBRA. 

for  the  priority  of  finger-counting  over  any  other.  Nearly 
every  system  of  numeral  notation  of  which  we  have  any 
knowledge  is  either  quinary,  decimal,  vigesimal,  or  a  mix- 
ture of  these  ;  *  that  is  to  say,  expresses  numbers  which  are 
greater  than  5  in  terms  of  5  and  lesser  numbers,  or  makes 
a  similar  use  of  10  or  20.  These  systems  point  to  primi- 
tive methods  of  reckoning  with  the  fingers  of  one  hand,  the 
fingers  of  both  hands,  all  the  fingers  and  toes,  respectively. 
Finger-counting,  furthermore,  is  universal  among  uncivil- 
ized tribes  of  the  present  day,  even  those  not  far  enough 
developed  to  have  numeral  words  beyond  2  or  3  represent- 
ing higher  numbers  by  holding  up  the  appropriate  number 
of  finger s.f 

J  83.  Spoken  Symbols.  Numeral  words  —  spoken  symbols 
—  would  naturally  arise  much  later  than  gesture  symbols. 
Wherever  the  origin  of  such  a  word  can  be  traced,  it  is 
found  to  be  either  descriptive  of  the  corresponding  finger 
symbol  or  —  when  there  is  nothing  characteristic  enough 
about  the  finger  symbol  to  suggest  a  word,  as  is  particularly 
the  case  with  the  smaller  numbers  —  the  name  of  some 
familiar  group  of  things.  Thus  in  the  languages  of  numer- 
ous tribes  the  numeral  5  is  simply  the  word  for  hand,  10 

*  Pure  quinary  and  vigesimal  systems  are  rare,  if  indeed  they  occur 
at  all.  As  an  example  of  the  former,  Tylor  (Primitive  Culture,  I, 
p.  261)  instances  a  Polynesian  number  series  which  runs  1,  2,  3,  4, 
5,  5-1,  5-2,  ...;  and  as  an  example  of  the  latter,  Cantor  (Geschichte 
der  Mathematik,  p.  8) ,  following  Pott,  cites  the  notation  of  the  Mayas 
of  Yucatan  who  have  special  words  for  20,  400,  8000,  160,000.  The 
Hebrew  notation,  like  the  Indo- Arabic,  affords  an  example  of  a  pure 
decimal  notation.  Mixed  systems  are  common.  Thus  the  Roman  is 
mixed  decimal  and  quinary,  the  Aztec  mixed  vigesimal  and  quinary. 
Speaking  generally,  the  quinary  and  vigesimal  systems  are  more  fre- 
quent among  the  lower  races,  the  decimal  among  the  higher.  (Primi- 
tive Culture,  I,  p.  262.) 

t  So,  for  instance,  the  aborigines  of  Victoria  and  the  Bororos  of 
Brazil  (Primitive  Culture,  I,  p.  244). 


PRIMITIVE  NUMERALS.  83 

for  both  hands,  20  for  "an  entire  mail"  (hands  and  feet)  ; 
while  2  is  the  word  for  the  eyes,  the  ears,  or  wings. # 

As  its  original  meaning  is  a  distinct  encumbrance  to  such 
a  word  in  its  use  as  a  numeral,  it  is  not  surprising  that  the 
numeral  words  of  the  highly  developed  languages  have  been 
so  modified  that  it  is  for  the  most  part  impossible  to  trace 
their  origin. 

The  practice  of  counting  with  numeral  words  probably 
arose  much  later  than  the  words  themselves.  There  is  an 
artificial  element  in  this  sort  of  counting  which  does  not 
appertain  to  primitive  counting  f  (see  §  5). 

One  fact  is  worth  reiterating  with  reference  to  both  the 
primitive  gesture  symbols  and  word  symbols  for  numbers. 
There  is  nothing  in  either  symbol  to  represent  the  indi- 
vidual characteristics  of  the  things  counted  or  their  arrange- 
ment. The  use  of  such  symbols,  therefore,  presupposes  a 
conviction  that  the  number  of  things  in  a  group  does  not 


*  In  the  language  of  the  Tamanacs  on  the  Orinoco  the  word  for  5 
means  "a  whole  hand,"  the  word  for  6,  "one  of  the  other  hand," 
and  so  on  up  to  9 ;  the  word  for  10  means  "both  hands,"  11,  "one 
to  the  foot,"  and  so  on  up  to  14;  15  is  "a  whole  foot,"  16,  "one  to 
the  other  foot,"  and  so  on  up  to  19;  20  is  "one  Indian,"  40,  "two 
Indians,"  etc.  Other  languages  rich  in  digit  numerals  are  the  Cayriri, 
Tupi,  Abipone,  and  Carib  of  South  America ;  the  Eskimo,  Aztec,  and 
Zulu  (Primitive  Culture,  I,  p.  247). 

"  Two  "  in  Chinese  is  a  word  meaning  "  ears,"  in  Thibet  "wing,"  in 
Hottentot  "hand."  (Gow,  Short  History  of  Greek  Mathematics,  p.  7.) 
See  also  Primitive  Culture,  I,  pp.  252-259. 

t  Were  there  any  reason  for  supposing  that  primitive  counting  was 
done  with  numeral  words,  it  would  be  probable  that  the  ordinals,  not ' 
the  cardinals,  were  the  earliest  numerals.     For  the  normal  order  of 
the  cardinals  must  have  been  fully  recognized  before  they  could  be 
used  in  counting. 

In  this  connection,  see  Kronecker,  Ueber  den  Zahlbegriff;  Jour- 
nal fur  die  reine  und  angewandte  Mathematik,  Vol.  101,  p.  337. 
Kronecker  goes  so  far  as  to  declare  that  he  finds  in  the  ordinal  num- 
bers the  natural  point  of  departure  for  the  development  of  the  number 
concept. 


84  NUMBER-SYSTEM  OF  ALGEBRA. 

depend  on  the  character  of  the  things  themselves  or  on 
their  collocation,  but  solely  on  their  maintaining  their  sepa- 
rateness  and  integrity. 

84.  Written  Symbols.  The  earliest  written  symbols  for 
number  would  naturally  be  mere  groups  of  strokes  —  I,  II, 
III,  etc.  Such  symbols  have  a  double  advantage  over 
gesture  symbols :  they  can  be  made  permanent,  and  are 
capable  of  indefinite  extension  —  there  being,  of  course,  no 
limit  to  the  numbers  of  strokes  which  may  be  drawn. 


II.     HISTOKIG   SYSTEMS   OP  NOTATION. 

85.  Egyptian  and  Phoenician.  This  written  symbolism 
did  not  take  on  the  systemless  character  it  must  have 
had,  had  counting  with  written  strokes  and  not  with  the 
fingers  been  the  primitive  method.  Perhaps  the  written 
strokes  were  employed  in  connection  with  counting  num- 
bers higher  than  10  on  the  fingers  to  indicate  how  often  all 
the  fingers  had  been  used ;  or  if  each  stroke  corresponded  to 
an  individual  in  the  group  counted,  they  were  arranged  as 
they  were  drawn  in  groups  of  10,  so  that  the  number  was 
represented  by  the  number  of  these  complete  groups  and 
the  strokes  in  a  remaining  group  of  less  than  10. 

At  all  events,  the  decimal  idea  very  early  found  expres- 
sion in  special  symbols  for  10, 100,  and  if  need  be,  of  higher 
powers  of  10.  Such  signs  are  already  at  hand  in  the  earliest 
known  writings  of  the  Egyptians  and  Phoenicians  in  which 
numbers  are  represented  by  unit  strokes  and  the  signs  for 
10,  100,  1000,  10,000,  and  even  100,000,  each  repeated  up  to 
9  times. 

86.  Greek.  In  two  of  the  best  known  notations  of 
antiquity,  the  old  Greek  notation  —  called  sometimes  the 


HISTORIC  SYSTEMS   OF  NOTATION.  85 

Herodianic,  sometimes  the  Attic  —  and  the  Roman,  a  primi- 
tive system  of  counting  on  the  fingers  of  a  single  hand  has 
left  its  impress  in  special  symbols  for  5. 

In  the  Herodianic  notation  the  only  symbols  —  apart 
from  certain  abbreviations  for  products  of  5  by  the  powers 
of  10  —  are  I,  r  (iwrc,  5),  A  (&*a,  10),  H  (eKaroV,  100), 
X  (x^Atot,  1000),  M  (fjLvptoi,  10,000) ;  all  of  them,  except  I,  it 
will  be  noticed,  initial  letters  of  numeral  words.  This  is 
the  only  notation,  it  may  be  added,  found  in  any  Attic 
inscription  of  a  date  before  Christ.  The  later  and,  for  the 
purposes  of  arithmetic,  much  inferior  notation,  in  which 
the  24  letters  of  the  Greek  alphabet  with  three  inserted 
strange  letters  represent  in  order  the  numbers  1,  2,  •••  10, 
20,  •••  100,  200,  •••  900,  was  apparently  first  employed  in 
Alexandria  early  in  the  3d  century  B.C.,  and  probably 
originated  in  that  city. 

87.  Roman.  The  Roman  notation  is  probably  of  Etrus- 
can origin.  It  has  one  very  distinctive  peculiarity :  the 
subtractive  meaning  of  a  symbol  of  lesser  value  when  it  pre- 
cedes one  of  greater  value,  as  in  IV  =  4  and  in  early  in- 
scriptions I IX  =  8.  In  nearly  every  other  known  system  of 
notation  the  principle  is  recognized  that  the  symbol  of  lesser 
value  shall  follow  that  of  greater  value  and  be  added  to  it. 

In  this  connection  it  is  worth  noticing  that  two  of  the 
four  fundamental  operations  of  arithmetic  —  addition  and 
multiplication  —  are  involved  in  the  very  use  of  special 
symbols  for  10  and  100,  for  the  one  is  but  a  symbol  for  the 
sum  of  10  units,  the  other  a  symbol  for  10  sums  of  10  units 
each,  or  for  the  product  10  x  10.  Indeed,  addition  is  prima- 
rily only  abbreviated  counting ;  multiplication,  abbreviated 
addition.  The  representation  of  a  number  in  terms  of  tens 
and  units,  moreover,  involves  the  expression  of  the  result 
of  a  division  (by  10)  in  the  number  of  its  tens  and  the 
result  of  a  subtraction  in  the  number  of  its  units.  It  does 
not  follow,  of  course,  that  the  inventors  of  the  notation  had 


86  NUMBER-SYSTEM  OF  ALGEBRA. 

any  such,  notion  of  its  meaning  or  that  these  inverse  opera- 
tions are,  like  addition  and  multiplication,  as  old  as  the 
symbolism  itself.  Yet  the  Etrusco-Roman  notation  testifies 
to  the  very  respectable  antiquity  of  one  of  them,  subtraction. 

88.  Indo- Arabic.  Associated  thus  intimately  with  the 
four  fundamental  operations  of  arithmetic,  the  character 
of  the  numeral  notation  determines  the  simplicity  or  com- 
plexity of  all  reckonings  with  numbers.  An  unusual  in- 
terest, therefore,  attaches  to  the  origin  of  the  beautifully 
clear  and  simple  notation  which  we  are  fortunate  enough 
to  possess.  What  a  boon  that  notation  is  will  be  appreci- 
ated by  one  who  attempts  an  exercise  in  division  with  the 
Eoman  or,  worst  of  all,  with  the  later  Greek  numerals. 

The  system  of  notation  in  current  use  to-day  may  be 
characterized  as  the  positional  decimal  system.  A  number 
is  resolved  into  the  sum  : 

an10w  +  an^10n-1  +  ...  +  a.10  +  a0, 

where  10"  is  the  highest  power  of  10  which  it  contains,  and 
an>  an-i>  '•'  ao  are  a^l  numbers  less  than  10 ;  and  then  repre- 
sented by  the  mere  sequence  of  numbers  anan_!  •  •  •  a0  —  it 
being  left  to  the  position  of  any  number  a{  in  this  sequence 
to  indicate  the  power  of  10  with  which  it  is  to  be  associ- 
ated. For  a  system  of  this  sort  to  be  complete  —  to  be 
capable  of  representing  all  numbers  unambiguously — a  sym- 
bol (0),  which  will  indicate  the  absence  of  any  particular 

power  of  10  from  the  sum  an10n  +  a^lO*""1  -\ J-%10  +  a0', 

is  indispensable.  Thus  without  0,  101  and  11  must  both  be 
written  11.  But  this  symbol  at  hand,  any  number  may  be 
expressed  unambiguously  in  terms  of  it  and  symbols  for 
12   ...  9 

The  positional  idea  is  very  old.  The  ancient  Babylonians 
commonly  employed  a  decimal  notation  similar  to  that  of 
the  Egyptians ;  but  their  astronomers  had  besides  this  a 
very  remarkable  notation,  a  sexagesimal  positional  system. 


HISTORIC  SYSTEMS   OF  NOTATION.  bi 

In  1854  a  brick  tablet  was  found  near  Senkereh  on  the 
Euphrates,  certainly  older  than  1600  B.C.,  on  one  face  of 
which  is  impressed  a  table  of  the  squares,  on  the  other,  a 
table  of  the  cubes  of  the  numbers  from  1  to  60.  The  squares 
of  1,  2,  •••  7  are  written  in  the  ordinary  decimal  notation, 
but  82,  or  64,  the  first  number  in  the  table  greater  than  60, 
is  written  1,  4  (1  X  60  +  4)  ;  similarly  92,  and  so  on  to  592, 
which  is  written  5S,  1  (58  X  60  + 1)  ;  while  602  is  written  1. 
The  same  notation  is  followed  in  the  table  of  cubes,  and  on 
other  tablets  which  have  since  been  found.  This  is  a  posi- 
tional system,  and  it  only  lacks  a  symbol  for  0  of  being  a 
perfect  positional  system. 

The  inventors  of  the  0-symbol  and  the  modern  complete 
decimal  positional  system  of  notation  were  the  Indians,  a 
race  of  the  finest  arithmetical  gifts. 

The  earlier  Indian  notation  is  decimal  but  not  positional. 
It  has  characters  for  10,  100,  etc.,  as  well  as  for  1,  2,  •••  9, 
and,  on  the  other  hand,  no  0. 

Most  of  the  Indian  characters  have  been  traced  back  to 
an  old  alphabet #  in  use  in  Northern  India  200  b.c  The 
original  of  each  numeral  symbol  4,  5,  6,  7,  8  (?),  9,  is  the 
initial  letter  in  this  alphabet  of  the  corresponding  numeral 
word  (see  table  on  page  89, f  colunfti  1).  The  characters 
first  occur  as  numeral  signs  in  certain  inscriptions  which  are 
assigned  to  the  1st  and  2d  centuries  a.d.  (column  2  of  table). 
Later  they  took  the  forms  given  in  column  3  of  the  table. 

When  0  was  invented  and  the  positional  notation  replaced 
the  old  notation   cannot  be  exactly  determined.      It  was 

*  Dr.  Isaac  Taylor,  in  his  book  "  The  Alphabet,"  names  this  alpha- 
bet the  Indo-Bactrian.  Its  earliest  and  most  important  monument  is 
the  version  of  the  edicts  of  King  Asoka  at  Kapur-di-giri.  In  this 
inscription,  it  may  be  added,  numerals  are  denoted  by  strokes,  as  I,  II, 
III,  MM,  Mill. 

t  Columns  1-5,  7,  8  of  the  table  on  page  89  are  taken  from  Tay- 
lor's Alphabet,  II,  p.  266  ;  column  6,  from  Cantor's  Geschichte  der 
Mathematik. 


88  NUMBER-SYSTEM  OF  ALGEBRA. 

certainly  later  than  400  a.d.,  and  there  is  no  evidence  that 
it  was  earlier  than  500  a.d.  The  earliest  known  instance 
of  a  date  written  in  the  new  notation  is  738  a.d.  By  the 
time  that  0  came  in,  the  other  characters  had  developed 
into  the  so-called  Devanagari  numerals  (table,  column  4), 
the  classical  numerals  of  the  Indians. 

The  perfected  Indian  system  probably  passed  over  to  the 
Arabians  in  773  a.d.,  along  with  certain  astronomical  writings. 
However  that  may  be,  it  was  expounded  in  the  early  part  of 
the  9th  century  by  Alkhwarizml,  and  from  that  time  on  spread 
gradually  throughout  the  Arabian  world,  the  numerals  tak- 
ing different  forms  in  the  East  and  in  the  West. 

Europe  in  turn  derived  the  system  from  the  Arabians  in 
the  12th  century,  the  "  Gobar  "  numerals  (table,  column  5) 
of  the  Arabians  of  Spain  being  the  pattern  forms  of  the  Euro- 
pean numerals  (table,  column  7).  The  arithmetic  founded 
on  the  new  system  was  at  first  called  algorithm  (after 
Alkhwarizmi),  to  distinguish  it  from  the  arithmetic  of  the 
abacus  which  it  came  to  replace. 

A  word  must  be  said  with  reference  to  this  arithmetic  on 
the  abacus.  In  the  primitive  abacus,  or  reckoning  table, 
unit  counters  were  used,  and  a  number  represented  by  the 
appropriate  number  of  these  counters  in  the  appropriate 
columns  of  the  instrument ;  e.g.  321  by  3  counters  in  the 
column  of  100' s,  2  in  the  column  of  10' s,  and  1  in  the 
column  of  units.  The  Eomans  employed  such  an  abacus 
in  all  but  the  most  elementary  reckonings,  it  was  in  use 
in  Greece,  and  is  in  use  to-day  in  China. 

Before  the  introduction  of  algorithm,  however,  reckon- 
ing on  the  abacus  had  been  improved  by  the  use  in  its 
columns  of  separate  characters  (called  apices)  for  each  of 
the  numbers  1,  2,  •  •  •  9,  instead  of  the  primitive  unit  counters. 
This  improved  abacus  reckoning  was  probably  invented  by 
Gerbert  (Pope  Sylvester  II.),  and  certainly  used  by  him 
at  Kheims  about  970-980,  and  became  generally  known  in 
the  following  century. 


HISTORIC  SYSTEMS  OF  DOTATION. 


89 


TABLE 

SHOWING  THE  EVOLUTION  OF  THE 

ARABIC 

CIPHERS. 

z 

E    2  H 

i-  a  r. 

INDIAN 

ARABIC 

EURO  PEA  N 

A.  D. 

GOBAR 

APICES 

LETTERS  OF 

INDO-BACT 

ALPHABE 

Sec.  I. 

Sec.  V. 

Sec.X. 

Sec.X. 

SecX. 

Sec.XII. 

Sec.XIV. 



r*\ 

1 

i 

1 

V 

1 

— 

0^-* 

\ 

I 

IT 

I 

7* 

B.  C. 

Sec.  II. 

— - 

*-~*~* 

£ 
a 

? 

5 

? 

V 

¥ 

$ 

f^ 

A 

4 

h 

h 

Y 

<\ 

i 

H 

< 

7 

y 

4 

c, 

s 

i 

Eb 

<? 

</ 

7 

? 

7 

7 

•* 

1 

7 

I 

6} 

< 

9 

8 

4 

8 

1 

? 

r 

9 

D 

3 

? 

O 

• 

^ 

o 

90  NUMBER-SYSTEM  OF  ALGEBRA. 

Now  these  apices  are  not  Boman  numerals,  but  symbols 
which  do  not  differ  greatly  from  the  Gobar  numerals  and 
are  clearly,  like  them,  of  Indian  origin.  In  the  absence  of 
positive  evidence  a  great  controversy  has  sprung  up  among 
historians  of  mathematics  over  the  immediate  origin  of  the 
apices.  The  only  earlier  mention  of  them  occurs  in  a  pas- 
sage of  the  geometry  of  Boetius,  which,  if  genuine,  was 
written  about  500  a.d.  Basing  his  argument  on  this  pas- 
sage, the  historian  Cantor  urges  that  the  earlier  Indian 
numerals  found  their  way  to  Alexandria  before  her  inter- 
course with  the  East' was  broken  off,  that  is,  before  the  end 
of  the  4th  century,  and  were  transformed  by  Boetius  into 
the  apices.  On  the  other  hand,  the  passage  in  Boetius  is 
quite  generally  believed  to  be  spurious,  and  it  is  maintained 
that  Gerbert  got  his  apices  directly  or  indirectly  from  the 
Arabians  of  Spain,  not  taking  the  0,  either  because  he  did 
not  learn  of  it,  or  because,  being  an  abacist,  he  did  not 
appreciate  its  value. 

At  all  events,  it  is  certain  that  the  Indo- Arabic  numerals, 
1,  2,  •••  9  (not  0),  appeared  in  Christian  Europe  more  than  a 
century  before  the  complete  positional  system  and  algorithm. 

The  Indians  are  the  inventors  not  only  of  the  positional 
decimal  system  itself,  but  of  most  of  the  processes  involved 
in  elementary  reckoning  with  the  system.  Addition  and 
subtraction  they  performed  quite  as  they  are  performed 
nowadays ;  multiplication  they  effected  in  many  ways,  ours 
among  them,  but  division  cumbrously. 


HI.     THE  FKA0TI0N. 

89.  Primitive  Fractions.  Of  the  artificial  forms  of  num- 
ber —  as  we  may  call  the  fraction,  the  irrational,  the  nega- 
tive, and  the  imaginary  in  contradistinction  to  the  positive 


THE  FRACTION.  91 

integer  —  all  but  the  fraction  are  creations  of  the  mathe- 
maticians. They  were  devised  to  meet  purely  mathematical 
rather  than  practical  needs.  The  fraction,  on  the  other 
hand,  is  already  present  in  the  oldest  numerical  records  — 
those  of  Egypt  and  Babylonia  —  was  reckoned  with  by  the 
Romans,  who  were  no  mathematicians,  and  by  Greek  mer- 
chants long  before  Greek  mathematicians  would  tolerate  it 
in  arithmetic. 

The  primitive  fraction  was  a  concrete  thing,  merely  an 
aliquot  part  of  some  larger  thing.  When  a  unit  of  measure 
was  found  too  large  for  certain  uses,  it  was  subdivided,  and 
one  of  these  subdivisions,  generally  with  a  name  of  its  own, 
made  a  new  unit.  Thus  there  arose  fractional  units  of 
measure,  and  in  like  manner  fractional  coins. 

In  time  the  relation  of  the  sub-unit  to  the  corresponding 
principal  unit  came  to  be  abstracted  with  greater  or  less 
completeness  from  the  particular  kind  of  things  to  which 
the  units  belonged,  and  was  recognized  when  existing  be- 
tween things  of  other  kinds.  The  relation  was  generalized, 
and  a  pure  numerical  expression  found  for  it. 

90.  Roman  Fractions.  Sometimes,  however,  the  relation 
was  never  completely  enough  separated  from  the  sub-units 
in  which  it  was  first  recognized  to  be  generalized.  The 
Romans,  for  instance,  never  got  beyond  expressing  all  their 
fractions  in  terms  of  the  uncia,  sicilicus,  etc.,  names  origin- 
ally of  subdivisions  of  the  old  unit  coin,  the  as. 

91.  Egyptian  Fractions.  Races  of  better  mathematical 
endowments  than  the  Romans,  however,  had  sufficient 
appreciation  of  the  fractional  relation  to  generalize  it  and 
give  it  an  arithmetical  symbolism. 

The  ancient  Egyptians  had  a  very  complete  symbolism 
of  this  sort.  They  represented  any  fraction  whose  numer- 
ator is  1  by  the  denominator  simply,  written  as  an  integer 
with  a  dot  over  it,  and  resolved  all  other  fractions  into 


92  NUMBER-SYSTEM  OF  ALGEBBA. 

sums  of  such  unit  fractions.  The  oldest  mathematical 
treatise  known,  —  a  papyrus  #  roll  entitled  "  Directions  for 
Attaining  to  the  Knowledge  of  All  Dark  Things/'  written 
by  a  scribe  named  Ahmes  in  the  reign  of  Ra-a-us  (therefore 
before  1700  B.C.),  after  the  model,  as  he  says,  of  a  more 
ancient  work,  —  opens  with  a  table  which  expresses  in  this ' 
manner  the  quotient  of  2  by  each  odd  number  from  5  to  99. 
Thus  the  quotient  of  2  by  5  is  written  3  15,  by  which  is 

meant   -  +  — ;  and  the   quotient   of  2   by  13,   8  52  104. 
o       JLo 

2 
Only   -,   among   the    fractions  having   numerators  which 
o 

differ  from  1,  gets  recognition  as  a  distinct  fraction  and 

receives  a  symbol  of  its  own. 

92.  Babylonian  or  Sexagesimal  Fractions.  The  fractional 
notation  of  the  Babylonian  astronomers  is  of  great  interest 
intrinsically  and  historically.  Like  their  notation  of  in- 
tegers it  is  a  sexagesimal  positional  notation.  The  denomi- 
nator is  always  60  or  some  power  of  60  indicated  by  the 
position  of  the  numerator,  which  alone  is  written.     (Jhe  f*> 

3  .p™  :~«4-«„^  «,v:»v»  ;« „i  4-~  22      30 

60  602' 
this  notation  be  written  22  30.  Thus  the  ability  to  represent 
fractions  by  a  single  integer  or  a  sequence  of  integers,  which 
the  Egyptians  secured  by  the  use  of  fractions  having  a  com- 
mon numerator,  1,  the  Babylonians  found  in  fractions  having 
common  denominators  and  the  principle  of  position.  The 
Egyptian  system  is  superior  in  that  it  gives  an  exact  expres- 
sion of  every  quotient,  which  the  Babylonian  can  in  general 
do  only  approximately.  As  regards  practical  usefulness, 
however,  the  Babylonian  is  beyond  comparison  the  better 
system.     Supply  the  0-symbol  and  substitute  10  for  60,  and 

*  The  Khind  papyrus  of  the  British  Museum ;  translated  by  A. 
Eisenlohr,  Leipzig,  1877. 


fraction  -,  for  instance,  which  is  equal  to 1 ,  would  in 


THE  FRACTION.  93 

this  notation  becomes  that  of  the  modern  decimal  fraction, 
in  whose  distinctive  merits  it  thus  shares. 

As  in  their  origin,  so  also  in  their  subsequent  history,* 
the  sexagesimal  fractions  are  intimately  associated  with 
astronomy.  The  astronomers  of  Greece,  India,  and  Arabia 
all  employ  them  in  reckonings  of  any  complexity,  in  those 
involving  the  lengths  of  lines  as  well  as  in  those  involving 
the  measures  of  angles.  So  the  Greek  astronomer,  Ptolemy 
(150  a.d.),  in  the  Almagest  (/meydXr}  owra&s)  measures  chords 
as  well  as  arcs  in  degrees,  minutes,  and  seconds  —  the  de- 
gree of  chord  being  the  60th  part  of  the  radius  as  the  degree 
of  arc  is  the  60th  part  of  the  arc  subtended  by  a  chord  equal 
to  the  radius. 

The  sexagesimal  fraction  held  its  own  as  the  fraction  par 
excellence  for  scientific  computation  until  the  16th  century, 
when  it  was  displaced  by  the  decimal  fraction  in  all  uses 
except  the  measurement  of  angles. 

93.  Greek  Fractions.  Fractions  occur  in  Greek  writings  — 
both  mathematical  and  non-mathematical — much  earlier  than 
Ptolemy,  but  not  in  arithmetic*  The  Greeks  drew  as  sharp 
a  distinction  between  pure  arithmetic,  dptO^TiKr),  and  the  art 
of  reckoning,  XoyiaTtKr),  as  between  pure  and  metrical  geom- 
etry. The  fraction  was  relegated  to  Xoyio-TLKr}.  There  is  no 
place  in  a  pure  science  for  artificial  concepts,  no  place,  there- 
fore, for  the  fraction  in  aptO/jLVTiKr} ;  such  was  the  Greek  posi- 
tion. Thus,  while  the  metrical  geometers  —  as  Archimedes 
(250  B.C.),  in  his  "Measure  of  the  Circle"  (kvkXov  /xerp^o-ts), 
and  Hero  (120  b.c.)  — employ  fractions,  neither  of  the  trea- 
tises on  Greek  arithmetic  before  Diophantus  (300  a.d.)  which 

*  The  usual  method  of  expressing  fractions  was  to  write  the  numer- 
ator with  an  accent,  and  after  it  the  denominator  twice  with  a  double 

17 
accent :  e.q.  iC  ku"  tea"  —  — -.     Before  sexagesimal  fractions  came  into 
j    %  21  ° 

vogue  actual  reckonings  with  fractions  were  effected  by  unit  fractions, 

of  which  only  the  denominators  (doubly  accented)  were  written. 


94  NUMBER-SYSTEM  OF  ALGEBRA. 

have  come  down  to  us  —  the  7th,  8th,  9th  books  of  Euclid's 
"Elements"  (300  B.C.),  and  the  " Introduction  to  Arithmetic '^ 
(etcraycoy^  apiOfjLrjTLKrj)  of  Mcomachus  (100  a.d.)  — recognizes 
the  fraction.  They  do,  it  is  true,  recognize  the  fractional 
relation.  Euclid,  for  instance,  expressly  declares  that  any 
number  is  either  a  multiple,  a  part,  or  parts  (/xep^),  i.e. 
multiple  of  a  part,  of  every  other  number  (Euc.  VII,  4), 
and  he  demonstrates  such  theorems  as  these : 

If  A  be  the  same  parts  of  B  that  C  is  of  D,  then  the  sum  or 
difference  of  A  and  C  is  the  same  parts  of  the  sum  or  difference 
of  B  and  D  that  A  is  of  B  (VII,  6  and  8). 

If  Abe  the  same  parts  of  B  that  C  is  of  D,  then,  alternately, 
A  is  the  same  parts  of  C  that  B  is  of '  D  (VII,  10). 

But  the  relation  is  expressed  by  two  integers,  that  which 
indicates  the  part  and  that  which  indicates  the  multiple. 
It  is  a  ratio,  and  Euclid  has  no  more  thought  of  expressing 
it  except  by  two  numbers  than  he  has  of  expressing  the 
ratio  of  two  geometric  magnitudes  except  by  two  magni- 
tudes. There  is  no  conception  of  a  single  number,  the 
fraction  proper,  the  quotient  of  one  of  these  integers  by 
the  other. 

In  the  apiOfxyrtKOL  of  Diophantus,  on  the  other  hand,  the 
last  and  transcendently  the  greatest  achievement  of  the 
Greeks  in  the  science  of  number,  the  fraction  is  granted 
the  position  in  elementary  arithmetic  which  it  has  held 
ever  since. 


IV.     OEIGIrT   OP   THE  IEEATIOrTAL. 

94.  The  Discovery  of  Irrational  Lines.  The  Greeks  attrib- 
uted the  discovery  of  the  Irrational  to  the  mathematician 
and  philosopher  Pythagoras  #  (525  B.C.). 

*  This  is  the  explicit  declaration  of  the  most  reliable  document  extant 
on  the  history  of  geometry  before  Euclid,  a  chronicle  of  the  ancient 


ORIGIN  OF  THE  IRRATIONAL.  95 

If,  as  is  altogether  probable, #  the  most  famous  theorem 
of  Pythagoras  —  that  the  square  on  the  hypothenuse  of  a 
right   triangle   is   equal   to   the   sum   of  the   squares   on   the 

geometers  which  Proems  (a.d.  450)  gives  in  his  commentary  on  Euclid, 
deriving  it  from  a  history  written  by  Eudemus  about  330  b.c.  This 
chronicle  credits  the  Egyptians  with  the  discovery  of  geometry  and 
Thales  (600  b.c.)  with  having  first  introduced  this  study  into  Greece. 

Thales  and  Pythagoras  are  the  founders  of  the  Greek  mathematics. 
But  while  Thales  should  doubtless  be  credited  with  the  first  conception 
of  an  abstract  deductive  geometry  in  contradistinction  to  the  practical 
empirical  geometry  of  Egypt,  the  glory  of  realizing  this  conception 
belongs  chiefly  to  Pythagoras  and  his  disciples  in  the  Greek  cities  of 
Italy  (Magna  Grsecia)  ;  for  they  established  the  principal  theorems 
respecting  rectilineal  figures.  To  the  Pythagoreans  the  discovery  of 
many  of  the  elementary  properties  of  numbers  is  due,  as  well  as  the 
geometric  form  which  characterized  the  Greek  theory  of  numbers 
throughout  its  history. 

In  the  middle  of  the  fifth  century  before  Christ  Athens  became  the 
principal  centre  of  mathematical  activity.  There  Hippocrates  of  Chios 
(430  b.c.)  made  his  contributions  to  the  geometry  of  the  circle,  Plato 
(380  b.c)  to  geometric  method,  Theaetetus  (380  b.c)  to  the  doctrine 
of  incommensurable  magnitudes,  and  Eudoxus  (360  b.c)  to  the  theory 
of  proportion!     There  also  was  begun  the  study  of  the  conies. 

About  300  b.c  the  mathematical  centre  of  the  Greeks  shifted  to 
Alexandria,  where  it  remained. 

The  third  century  before  Christ  is  the  most  brilliant  period  in  Greek 
mathematics.  At  its  beginning  —  in  Alexandria  —  Euclid  lived  and 
taught  and  wrote  his  Elements,  collecting,  systematizing,  and  per- 
fecting the  work  of  his  predecessors.  Later  (about  250)  Archimedes 
of  Syracuse  flourished,  the  greatest  mathematician  of  antiquity  and 
founder  of  the  science  of  mechanics  ;  and  later  still  (about  230)  Apol- 
lonius  of  Perga,  "the  great  geometer,"  whose  Conies  marks  the  cul- 
mination of  Greek  geometry. 

Of  the  later  Greek  mathematicians,  besides  Hero  and  Diophantus, 
of  whom  an  account  is  given  in  the  text,  and  the  great  summarizer  of 
the  ancient  mathematics,  Pappus  (300  a.d.),  only  the  famous  astrono- 
mers Hipparchus  (130  b.c)  and  Ptolemy  (150  a.d.)  call  for  mention 
here.  To  them  belongs  the  invention  of  trigonometry  and  the  first 
trigonometric  tables,  tables  of  chords. 

The  dates  in  this  summary  are  from  Gow's  Hist,  of  Greek  Math. 

*  Compare  Cantor,  Geschichte  der  Mathematik,  p.  153. 


> 


96  NUMBER-SYSTEM  OF  ALGEBRA. 

other  two  sides  —  was  suggested  to  him  by  the  fact  that 
32  _j_  42  _  52^  in  connection  with  the  fact  that  the  triangle 
whose  sides  are  3,  4,  5,  is  right-angled, — for  both  almost 
certainly  fell  within  the  knowledge  of  the  Egyptians,  -7  he 
would  naturally  have  sought,  after  he  had  succeeded  In 
demonstrating  the  geometric  theorem  generally,  for  number 
triplets  corresponding  to  the  sides  of  any  right  triangle  as 
do  3,  4,  5  to  the  sides  of  the  particular  triangle. 

The  search  of  course  proved  fruitless,  fruitless  even  in 
the  case  which  is  geometrically  the  simplest,  that  of  the 
isosceles  right  triangle.  To  discover  that  it  was  necessarily 
fruitless ;  in  the  face  of  preconceived  ideas  and  the  appar- 
ent testimony  of  the  senses,  to  conceive  that  lines  may  exist 
which  have  no  common  unit  of  measure,  however  small  that 
unit  be  taken ;  to  demonstrate  that  the  hypothenuse  and 
side  of  the  isosceles  right  triangle  actually  are  such  a  pair 
of  lines,  was  the  great  achievement  of  Pythagoras. # 

95.   Consequences  of  this  Discovery  in  Greek  Mathematics. 

One  must  know  the  antecedents  and  follow  the  consequences 
of  this  discovery  to  realize  its  great  significance.     It  was 

*  His  demonstration  may  easily  have  been  the  following,  which  was 
old  enough  in  Aristotle's  time  (340,fe.c.). to  be  made  the  subject  of  a 
popular  reference,  and  which  is  to  be  found  at  the  end  of  the  10th  book 
in  all  old  editions  of  Euclid's  Elements : 

If  there  be  any  line  which  the  side  and  diagonal  of  a  square  both 
contain  an  exact  number  of  times,  let  their  lengths  in  terms  of  this 
line  be  a  and  b  respectively  ;  then  b2  =  2  a2. 

The  numbers  a  and  b  may  have  a  common  factor,  7  ;  so  that  a— ay 
and  6  =  #7,  where  a  and  /3  are  prime  to  each  other.  The  equation 
b2  =  2  a2  then  reduces,  on  the  removal  of  the  factor  72  common  to  both 
its  members,  to  fi2  =  2  a2. 

From  this  equation  it  follows  that  #2,  and  therefore  £,  is  an  even 
number,  and  hence  that  a  which  is  prime  to  #  is  odd. 

But  set  j8  =  2£',  where  /3'  is  integral,  in  the  equation  &2  =  2a2;  it 
becomes  4  &'2  —  2  a2,  or  2  &'2  =  a2,  whence  a2,  and  therefore  a,  is  even. 

a  has  thus  been  proven  to  be  both  odd  and  even,  and  is  therefore 
not  a  number. 


ORIGIN   OF  THE  IRRATIONAL.  97 

the  first  recognition  of  the  fundamental  difference  between 
the  geometric  magnitudes  and  number,  which  Aristotle  for- 
mulated brilliantly  200  years  later  in  his  famous  distinc- 
tion between  the  continuous  and  the  discrete,  and  as  such 
was  potent  in  bringing  about  that  complete  banishment  of 
numerical  reckoning  from  geometry  which  is  so  character- 
istic of  this  department  of  Greek  mathematics  in  its  best, 
its  creative  period. 

No  one  before  Pythagoras  had  questioned  the  possibility 
of  expressing  all  size  relations  among  lines  and  surfaces  in 
terms  of  number,  —  rational  number  of  course.  Indeed, 
except  that  it  recorded  a  few  facts  regarding  congruence  of 
figures  gathered  by  observation,  the  Egyptian  geometry  was 
nothing  else  than  a  meagre  collection  of  formulas  for  com- 
puting areas.     The  earliest  geometry  was  metrical. 

But  to  the  severely  logical  Greek  no  alternative  seemed 
possible,  when  once  it  was  known  that  lines  exist  whose 
lengths  —  whatever  unit  be  chosen  for  measuring  them  — 
cannot  both  be  integers,  than  to  have  done  with  number 
and  measurement  in  geometry  altogether.  Congruence  be- 
came not  only  the  final  but  the  sole  test  of  equality.  For 
the  study  of  size  relations  among  unequal  magnitudes  a 
pure  geometric  theory  of  proportion  was  created,  in  which 
proportion,  not  ratio,  was  the  primary  idea,  the  method  of 
exhaustions  making  the  theory  available  for  figures  bounded 
by  curved  lines  and  surfaces. 

The  outcome  was  the  system  of  geometry  which  Euclid 
expounds  in  his  Elements  and  of  which  Apollonius  makes 
splendid  use  in  his  Conies,  a  system  absolutely  free  from 
extraneous  concepts  or  methods,  yet,  within  its  limits,  of 
great  power. 

It  need  hardly  be  added  that  it  never  occurred  to  the 
Greeks  to  meet  the  difficulty  which  Pythagoras'  discovery 
had  brought  to  light  by  inventing  an  irrational  number, 
itself  incommensurable  with  rational  numbers.  For  arti- 
ficial concepts  such  as  that  they  had  neither  talent  nor  liking. 


98  NUMBER-SYSTEM  OF  ALGEBRA, 

On  the  other  hand,  they  did  develop  the  theory  of  irra- 
tional magnitudes  as  a  department  of  their  geometry,  the 
irrational  line,  surface,  or  solid  being  one  incommensurable 
with  some  chosen  (rational)  line,  surface,  solid.  Such  a 
theory  forms  the  content  of  the  most  elaborate  book  of 
Euclid's  Elements,  the  10th. 

96.  Approximate  Values  of  Irrationals.  In  the  practical 
or  metrical  geometry  which  grew  up  after  the  pure  geometry 
had  reached  its  culmination,  and  which  attained  in  the  works 
of  Hero  the  Surveyor  almost  the  proportions  of  our  modern 
elementary  mensuration,*  approximate  values  of  irrational 
numbers  played  a  very  important  r6le.  Nor  do  such  approx- 
imations appear  for  the  first  time  in  Hero.  In  Archimedes' 
"  Measure  of  the  Circle  "  a  number  of  excellent  approxima- 

-  .       22 
tions  occur,  among  them  the  famous  approximation  —  for 

7r,  the  ratio  of  the  circumference  of  a  circle  to  its  diam- 
eter.  The  approximation  -  for  V2  is  reputed  to  be  as  old 
as  Plato. 

It  is  not  certain  how  these  approximations  were  effected. f 
They  involve  the  use  of  some  method  for  extracting  square 
roots.  The  earliest  explicit  statement  of  the  method  in 
common  use  to-day  for  extracting  square  roots  of  numbers 
(whether  exactly  or  approximately)  occurs  in  the  com- 
mentary of  Theon  of  Alexandria  (380  a.d.)  on  Ptolemy's 


*  The  formula  Vs  (s  —  a)(s  —  b)  (s  —  c)  for  the  area  of  a  triangle  in 
terms  of  its  sides  is  due  to  Hero. 

t  Many  attempts  have  been  made  to  discover  the  methods  of 
approximation  used  by  Archimedes  and  Hero  from  an  examination 

of  their  results,  but  with  little  success.     The  formula  Va2  ±  b  =  a  ±  — ■ 

2a 

will  account  for  some  of  the  simpler  approximations,  but  no  single 
method  or  set  of  methods  have  been  found  which  will  account  for  the 
more  difficult.  See  Gunther:  Die  quadratischen  Irrationalitaten  der 
Alten  und  deren  Entwicldungsmethoden.  Leipzig,  1882.  Also  in 
Handbuch  der  klassischen  Altertums-Wissenschaft,  liter.    Halbband. 


GREEK  ALGEBRA.  99 

Almagest.  Theon,  who  like  Ptolemy  employs  sexagesimal 
fractions,  thus  finds  the  length  of  the  side  of  a  square  con- 
taining 4500°  to  be  67°  1'  55". 

97.  The  Later  History  of  the  Irrational  is  deferred  to  the 
chapters  which  follow  (§§  106,  108,  112,  121,  129). 

It  will  be  found  that  the  Indians  permitted  the  simplest 
forms  of  irrational  numbers,  surds,  in  their  algebra,  and  that 
they  were  followed  in  this  by  the  Arabians  and  the  mathema- 
ticians of  the  Kenaissance,  but  that  the  general  irrational 
did  not  make  its  way  into  algebra  until  after  Descartes. 


V.     OKIGIN  OF  THE  NEGATIVE  AND  THE  IMAGINAKY. 

THE  EQUATION. 

98.  The  Equation  in  Egyptian  Mathematics.  While  the 
irrational  originated  in  geometry,  the  negative  and  the 
imaginary  are  of  purely  algebraic  origin.  They  sprang 
directly  from  the  algebraic  equation. 

The  authentic  history  of  the  equation,  like  that  of  geome- 
try and  arithmetic,  begins  in  the  book  of  the  old  Egyptian 
scribe  Ahmes.  For  Ahmes,  quite  after  the  present  method, 
solves  numerical  problems  which  admit  of  statement  in  an 
equation  of  the  first  degree  involving  one  unknown  quantity.* 

99.  In  the  Earlier  Greek  Mathematics.  The  equation  was 
slow  in  arousing  the  interest  of  Greek  mathematicians.  They 
were  absorbed  in  geometry,  in  a  geometry  whose  methods 
were  essentially  non-algebraic. 

To  be  sure,  there  are  occasional  signs  of  a  concealed 
algebra  under  the  closely  drawn  geometric  cloak.     Euclid 

*  His  symbol  for  the  unknown  quantity  is  the  word  hau,  meaning 
heap. 


100  NUMBER-SYSTEM  OF  ALGEBRA. 

solves  three  geometric  problems  which,  stated  algebraically, 
are  but  the  three  forms  of  the  quadratic ;  or  +  ax  =  b2, 
x2  =  ax  +  b2,  x2  +  b2  =  ax*  And  the  Conies  of  Apollonius, 
so  astonishing  if  regarded  as  a  product  of  the  pure  geo- 
metric method  used  in  its  demonstrations,  when  stated  in 
the  language  of  algebra,  as  recently  it  has  been  stated  by 
Zeuthen,f  almost  convicts  its  author  of  the  use  of  algebra 
as  his  instrument  of  investigation. 

100.  Hero.  But  in  the  writings  of  Hero  of  Alexandria 
(120  b.c.)  the  equation  first  comes  clearly  into  the  light 
again.  Hero  was  a  man  of  practical  genius  whose  aim  was 
to  make  the  rich  pure  geometry  of  his  predecessors  available 
for  the  surveyor.  With  him  the  rigor  of  the  old  geometric 
method  is  relaxed ;  proportions,  even  equations,  among  the 
measures  of  magnitudes  are  permitted  where  the  earlier 
geometers  allow  only  proportions  among  the  magnitudes 
themselves ;  the  theorems  of  geometry  are  stated  metrically, 
in  formulas ;  and  more  than  all  this,  the  equation  becomes 
a  recognized  geometric  instrument. 

Hero  gives  for  the  diameter  of  a  circle  in  terms  of  s,  the 
sum  of  diameter,  circumference,  and  area,  the  formula :  t 

^_Vl54s-f-841-^-29 
a~       '      IT 

He  could  have  reached  this  formula  only  by  solving  a 
quadratic  equation,  and  that  not  geometrically,  —  the  nature 
of  the  oddly  constituted  quantity  s  precludes  that  suppo- 
sition,—  but  by  a  purely  algebraic  reckoning  like  the 
following  : 

7rd2 

The  area  of  a  circle  in  terms  of  its  diameter  being  — — , 

*  Elements,  VI,  29,  28  ;  Data,  84,  85. 

t  Die  Lehre  von  den  Kegelschnitten  im  Altertum.  Copenhagen, 
1886. 

\  See  Cantor  ;  Geschichte  der  Mathematik,  p.  341. 


GREEK  ALGEBRA.  101 

the   length  of  its  circumference   71-d,  and   tt   according  to 

22 
Archimedes'  approximation  — ,  we  have  the  equation : 

,  ,  wd2  ,      ,         11  #  ,  29  , 

s  =  d  -\ h  7ra,  or  -—d~-\ d  =  s. 

4  14  7 

Clearing  of  fractions,  multiplying  by  11,  and  completing 
the  square, 

121  d2  +  638  d  +  841  =  154  s  +  841, 


whence  11  d  +  29  =  Vlo4  s  +  841, 

Vl54s  +  841-29 


or  d  =  - 


11 


Except  that  he  lacked  an  algebraic  symbolism,  therefore, 
Hero  was  an  algebraist,  an  algebraist  of  power  enough  to 
solve  an  affected  quadratic  equation. 

101.  Diophantus  (300  a.d.  ?).  The  last  of  the  Greek 
mathematicians,  Diophantus  of  Alexandria,  was  a  great 
algebraist. 

The  period  between  him  and  Hero  was  not  rich  in  cre- 
ative mathematicians,  but  it  must  have  witnessed  a  grad- 
ual development  of  algebraic  ideas  and  of  an  algebraic 
symbolism. 

At  all  events,  in  the  apiOix-qriKa  of  Diophantus  the  alge- 
braic equation  has  been  supplied  with  a  symbol  for  the 
unknown  quantity,  its  powers  and  the  powers  of  its  recip- 
rocal to  the  6th,  and  a  symbol  for  equality.  Addition  is 
represented  by  mere  juxtaposition,  but  there  is  a  special 
symbol,  /p,  for  subtraction.  On  the  other  hand,  there  are 
no  general  symbols  for  known  quantities,  —  symbols  to 
serve  the  purpose  which  the  first  letters  of  the  alphabet 
are  made  to  serve  in  elementary  algebra  nowadays,  —  there- 
fore no  literal  coefficients  and  no  general  formulas. 

With  the  symbolism  had  grown  up  many  of  the  formal 
rules  of  algebraic  reckoning  also.     Diophantus  prefaces  the 


102  NUMBER-SYSTEM  OF  ALGEBRA. 

apL$fjLr)TiKd  with,  rules  for  trie  addition,  subtraction,  and  mul- 
tiplication of  polynomials.  He  states  expressly  that  the 
product  of  two  subtractive  terms  is  additive. 

The  apiOfjLrjTLKOL  itself  is  a  collection  of  problems  concern- 
ing numbers,  some  of  which  are  solved  by  determinate 
algebraic  equations,  some  by  indeterminate. 

Determinate  equations  are  solved  which  have  given  posi- 
tive integers  as  coefficients,  and  are  of  any  of  the  forms 
axm  =  bxn,  ax2  -f-  bx  —  c,  ax2  -f-  c  =  bx,  ax2  =  bx-\-c9  also  a 
single  cubic  equation,  xs  -f  x  =  Ax2  +  4.  In  reducing  equa- 
tions to  these  forms,  equal  quantities  in  opposite  members 
are  cancelled  and  subtractive  terms  in  either  member  are 
rendered  additive  by  transposition  to  the  other  member. 

The  indeterminate  equations  are  of  the  form  y2  =  ax2 
+  bx  -f  c,  Diophantus  regarding  any  pair  of  positive  rational 
numbers  (integers  or  fractions)  as  a  solution  which,  substi- 
tuted for  y  and  x,  satisfy  the  equation.*  These  equations  are 
handled  with  marvellous  dexterity  in  the  apiB^TiKo..  No 
effort  is  made  to  develop  general  comprehensive  methods, 
but  each  exercise  is  solved  by  some  clever  device  suggested 
by  its  individual  peculiarities.  Moreover,  the  discussion  is 
never  exhaustive,  one  solution  sufficing  when  the  possible 
number  is  infinite.  Yet  until  some  trace  of  indeterminate 
equations  earlier  than  the  apiOfxrjriKa  is  discovered,  Diophan- 
tus must  rank  as  the  originator  of  this  department  of 
mathematics. 

The  determinate  quadratic  is  solved  by  the  method  which 
we  have  already  seen  used  by  Hero.  The  equation  is  first 
multiplied  throughout  by  a  number  which  renders  the  co- 
efficient of  x2  a  perfect  square,  the  "  square  is  completed," 
the  square  root  of  both  members  of  the  equation  taken,  and 

*  The  designation  "  Diophantine  equations,"  commonly  applied  to 
indeterminate  equations  of  the  first  degree  when  investigated  for  inte- 
gral solutions,  is  a  striking  misnomer.  Diophantus  nowhere  considers 
such  equations,  and,  on  the  other  hand,  allows  fractional  solutions  of 
indeterminate  equations  of  the  second  degree. 


INDIAN  ALGEBRA.  103 

the  value  of  x  reckoned  out  from  the  result.     Thus  from 
ax2  -f-  c  =  bx  is  derived  first  the  equation 
a2x2  -f  ac  =  abx, 

then  a2x2  —  abx  +  (-)  =  f  x  J  —etc, 

then  ax  —  -  =  +  l(  -  J  —  ac, 


V(IT- 


and  finally,  x  = *■* 

The  solution  is  regarded  as  possible  only  when  the  num- 
ber under  the  radical  is  a  perfect  square  (it  must,  of  course, 
be  positive),  and  only  one  root  —  that  belonging  to  the 
positive  value  of  the  radical  —  is  ever  recognized. 

Thus  the  number  system  of  Diophantus  contained  only 
the  positive  integer  and  fraction ;  the  irrational  is  excluded ; 
and  as  for  the  negative,  there  is  no  evidence  that  a  Greek 
mathematician  ever  conceived  of  such  a  thing, — certainly 
not  Diophantus  with  his  three  classes  and  one  root  of 
affected  quadratics.  The  position  of  Diophantus  is  the 
more  interesting  in  that  in  the  dpitf/x^TiKa  the  Greek  science 
of  number  culminates. 

102.  The  Indian  Mathematics.  The  pre-eminence  in  math- 
ematics passed  from  the  Greeks  to  the  Indians.  Three 
mathematicians  of  India  stand  out  above  the  rest :  Arya- 
bhatta  (born  476  a.d.),  Brahmagupta  (born  598  a.d.),  Bhds- 
kara  (born  1114  a.d.).  While  all  are  in  the  first  instance 
astronomers,  their  treatises  also  contain  full  expositions  of 
the  mathematics  auxiliary  to  astronomy,  their  reckoning, 
algebra,  geometry,  and  trigonometry. # 

*  The  mathematical  chapters  of  Brahmagupta  and  Bhaskara  have 
been  translated  into  English  by  Colebrooke :  "Algebra,  Arithmetic, 
and  Mensuration,  from  the  Sanscrit  of  Brahmagupta  and  Bhaskara," 
1817  ;  those  of  Aryabhatta  into  French  by  L.  Kodet  (Journal  Asiatique, 
1879). 


104  NUMBER-SYSTEM  OF  ALGEBRA. 

An  examination  of  the  writings  of  these  mathematicians 
and  of  the  remaining  mathematical  literature  of  India  leaves 
little  room  for  doubt  that  the  Indian  geometry  was  taken 
bodily  from  Hero,  and  the  algebra  —  whatever  there  may 
have  been  of  it  before  Aryabhatta  —  at  least  powerfully 
affected  by  Diophantus.  Nor  is  there  occasion  for  surprise 
in  this.  Aryabhatta  lived  two  centuries  after  Diophantus 
and  six  after  Hero,  and  during  those  centuries  the  East  had 
frequent  communication  with  the  West  through  various 
channels.  In  particular,  from  Trajan's  reign  till  later  than 
300  a.d.  an  active  commerce  was  kept  up  between  India 
and  the  east  coast  of  Egypt  by  way  of  the  Indian  Ocean. 

Greek  geometry  and  Greek  algebra  met  very  different 
fates  in  India.  The  Indians  lacked  the  endowments  of  the 
geometer.  So  far  from  enriching  the  science  with  new 
discoveries,  they  seem  with  difficulty  to  have  kept  alive 
even  a  proper  understanding  of  Hero's  metrical  formulas. 
But  algebra  flourished  among  them  wonderfully.  Here  the 
fine  talent  for  reckoning  which  had  created  a  perfect  nu- 
meral notation,  supported  by  a  talent  equally  fine  for  sym- 
bolical reasoning,  found  a  great  opportunity  and  made 
great  achievements.  With  Diophantus  algebra  is  no  more 
than  an  art  by  which  disconnected  numerical  problems  are 
solved ;  in  India  it  rises  to  the  dignity  of  a  science,  with 
general  methods  and  concepts  of  its  own. 

103.  Its  Algebraic  Symbolism.  Eirst  of  all,  the  Indians 
devised  a  complete,  and  in  most  respects  adequate,  sym- 
bolism. Addition  was  represented,  as  by  Diophantus,  by 
mere  juxtaposition ;  subtraction,  exactly  as  addition,  except 
that  a  dot  was  written  over  the  coefficient  of  the  subtra- 
hend. The  syllable  bha  written  after  the  factors  indicated 
a  product ;  the  divisor  written  under  the  dividend,  a  quo- 
tient ;  a  syllable,  ka,  written  before  a  number,  its  (irrational) 
square  root ;  one  member  of  an  equation  placed  over  the 
other,  their  equality.     The  equation  was  also  provided  with 


INDIAN  ALGEBRA.  105 

symbols  for  any  number  of  unknown  quantities  and  their 
powers. 

104.  Its  Invention  of  the  Negative.  The  most  note- 
worthy feature  of  this  symbolism  is  its  representation  of 
subtraction.  To  remove  the  subtractive  symbol  from  be- 
tween minuend  and  subtrahend  (where  Diophantus  had 
placed  his  symbol  41),  to  attach  it  wholly  to  the  subtrahend 
and  thus  connect  this  modified  subtrahend  with  the  minuend 
additively,  is,  formally  considered,  to  transform  the  sub- 
traction of  a  positive  quantity  into  the  addition  of  the 
corresponding  negative.  It  suggests  what  other  evidence 
makes  certain,  that  algebra  owes  to  India  the  immensely 
useful  concept  of  the  absolute  negative. 

Thus  one  of  these  dotted  numbers  is  allowed  to  stand 
by  itself  as  a  member  of  an  equation.  Bhaskara  recognizes 
the  double  sign  of  the  square  root,  as  well  as  the  impossi- 
bility of  the  square  root  of  a  negative  number  (which  is 
very  interesting,  as  being  the  first  dictum  regarding  the  imag- 
inary), and  no  longer  ignores  either  root  of  the  quadratic. 
More  than  this,  recourse  is  had  to  the  same  expedients 
for  interpreting  the  negative,  for  attaching  a  concrete  phys- 
ical idea  to  it,  which  persist  to  this  day.  The  primary 
meaning  of  the  very  name,  given  the  negative  was  debt,  as 
that  given  the  positive  was  means.  The  opposition  between 
the  two  was  also  pictured  by  lines  described  in  opposite 
directions. 

105.  Its  Use  of  Zero.  But  the  contributions  of  the  Ind- 
ians to  the  fund  of  algebraic  concepts  did  not  stop  with  the 
absolute  negative. 

They  made  a  number  of  0^  and  though  some  of  their 
reckonings  with  it  are  childish,  Bhaskara,    at  least,   had 

sufficient  understanding  of  the  nature  of  the  "  quotient "  - 

(infinity)  to  say  "it  suffers  no  change,  however  much  it 
is  increased  or  diminished."     He  associates  it  with  Deity. 


106  NUMBEB-SYSTEM  OF  ALGEBBA. 

106.  Its  Use  of  Irrational  Numbers.  Again,  the  Indians 
were  the  first  to  reckon  with  irrational  square  roots  as  with 
numbers;  Bhaskara  extracting  square  roots  of  binomial 
surds  and  rationalizing  irrational  denominators  of  fractions 
even  when  these  are  polynomial.  Of  course  they  were  as 
little  able  rigorously  to  justify  such  a  procedure  as  the 
Greeks ;  less  able,  in  fact,  since  they  had  no  equivalent  of 
the  method  of  exhaustions.  But  it  probably  never  occurred 
to  them  that  justification  was  necessary;  they  seem  to  have 
been  unconscious  of  the  gulf  fixed  between  the  discrete  and 
continuous.  And  here,  as  in  the  case  of  0  and  the  negative, 
with  the  confidence  of  apt  and  successful  reckoners,  they 
were  ready  to  pass  immediately  from  numerical  to  purely 
symbolical  reasoning,  ready  to  trust  their  processes  even 
where  formal  demonstration  of  the  right  to  apply  them 
ceased  to  be  attainable.  Their  skill  was  too  great,  their 
instinct  too  true,  to  allow  them  to  go  far  wrong. 

107.  Determinate  and  Indeterminate  Equations  in  Indian 
Algebra.  As  regards  equations  —  the  only  changes  which 
the  Indian  algebraists  made  in  the  treatment  of  determinate 
equations  were  such  as  grew  out  of  the  use  of  the  negative. 
This  brought  the  triple  classification  of  the  quadratic  to  an 
end  and  secured  recognition  for^both  roots  of  the  quadratic. 

Brahmagupta  solves  the  quadratic  by  the  rule  of  Hero 
and  Diophantus,  of  which  he  gives  an  explicit  and  gen- 
eral statement.  Cridhara,  a  mathematician  of  some  dis- 
tinction belonging  to  the  period  between  Brahmagupta  and 
Bhaskara,  made  the  improvement  of  this  method  which 
consists  in  first  multiplying  the  equation  throughout  by 
four  times  the  coefficient  of  the  square  of  the  unknown 
quantity  and  so  preventing  the  occurrence  of  fractions 
under  the  radical  sign.* 

Bhaskara  also  solves  a  few  cubic  and  biquadratic  equa- 
tions by  special  devices. 

*  This  method  still  goes  under  the  name  "  Hindoo  method." 


ARABIAJS    ALGEBRA.  107 

The  theory  of  indeterminate  equations,  on  the  other  hand, 
made  great  progress  in  India.  The  achievements  of  the 
Indian  mathematicians  in  this  beautiful  but  difficult  depart- 
ment of  the  science  are  as  brilliant  as  those  of  the  Greeks 
in  geometry.  They  created  the  doctrine  of  the  indetermi- 
nate equation  of  the  first  degree,  ax -{-by  =  c,  which  they 
treated  for  integral  solutions  by  the  method  of  continued 
fractions  in  use  to-day.  They  worked  also  with  equations  of 
the  second  degree  of  the  forms  ax2  -\-b  =  cy2,  xy  =  ax  -{-by  -f-  c, 
originating  general  and  comprehensive  methods  where  Dio- 
phantus  had  been  content  with  clever  jugglery. 

108.  The  Arabian  Mathematics.  The  Arabians  were  the 
instructors  of  modern  Europe  in  the  ancient  mathematics. 
The  service  which  they  rendered  in  the  case  of  the  numeral 
notation  and  reckoning  of  India  they  rendered  also  in  the 
case  of  the  geometry,  algebra,  and  astronomy  of  the  Greeks 
and  Indians.  Their  own  contributions  to  mathematics  are 
unimportant.  Their  receptiveness  for  mathematical  ideas 
was  extraordinary,  but  they  had  little  originality. 

The  history  of  Arabian  mathematics  begins  with  the  reign 
of  Almansur  (754-775), #  the  second  of  the  Abbasid  caliphs. 

It  is  related  (by  Ibn-al-Adami,  about  900)  that  in  this 
reign,  in  the  year  773,  an  Indian  brought  to  Bagdad  certain 
astronomical  writings  of  his  country,  which  contained  a 
method  called  u  Sindhind,"  for  computing  the  motions  of 
the  stars,  —  probably  portions  of  the  Siddhanta  of  Brahma- 
gupta,  —  and  that  Alfazari  was  commissioned  by  the  caliph 
to  translate  them  into  Arabic. t     Inasmuch  as  the  Indian 


*  It  was  Almansur  who  transferred  the  throne  of  the  caliphs  from 
Damascus  to  Bagdad  which  immediately  became  not  only  the  capital 
city  of  Islam,  but  its  commercial  and  intellectual  centre. 

t  This  translation  remained  the  guide  of  the  Arabian  astronomers 
until  the  reign  of  Almamun  (813-833),  for  whom  Alkhwarizmi  pre- 
pared his  famous  astronomical  tables  (820).  Even  these  were  based 
chiefly  on  the  "Sindhind,"  though  some  of  the  determinations  were 
made  by  methods  of  the  Persians  and  Ptolemy. 


108  NUMBER-SYSl^ip^OF  ALGEBRA. 

astronomers  put  full  expositions  of  their  reckoning,  algebra, 
and  geometry  into  their  treatises,  Alfazarfs  translation  laid 
open  to  his  countrymen  a  rich  treasure  of  mathematical  ideas 
and  methods. 

It  is  impossible  to  set  a  date  to  the  entrance  of  Greek 
ideas.  They  must  have  made  themselves  felt  at  Damascus, 
the  residence  of  the  later  Omayyad  caliphs,  for  that  city  had 
numerous  inhabitants  of  Greek  origin  and  culture.  But  the 
first  translations  of  Greek  mathematical  writings  were  made 
in  the  reign  of  Harun  Arraschid  (786-809),  when  Euclid's 
Elements  and  Ptolemy's  Almagest  were  put  into  Arabic. 
Later  on,  translations  were  made  of  Archimedes,  Apollo- 
nius,  Hero,  and  last  of  all,  of  Diophantus  (by  Abu'l  Wafa, 
940-998). 

The  earliest  mathematical  author  of  the  Arabians  is 
Alkhwarizmi,  who  flourished  in  the  first  quarter  of  the  9th 
century.  Besides  astronomical  tables,  he  wrote  a  treatise 
on  algebra  and  one  on  reckoning  (elementary  arithmetic). 
The  latter  has  already  been  mentioned.  It  is  an  exposition 
of  the  positional  reckoning  of  India,  the  reckoning  which 
mediaeval  Europe  named  after  him  Algorithm. 
a J  ^he  treatise  on  algebra  bears  a  title  in  which  the  word 
Mlgebra^  appears  for  the  first  time :  viz.,  Ald/jebr  walmu- 
p  Jcdbala.  Aldjehr  {i.e.  reduction)  signifies  the  making  of  all 
terms  of  an  equation  positive  by  transferring  negative  terms 
to  the  opposite  member  of  the  equation;  almukabala  (i.e. 
opposition),  the  cancelling  of  equal  terms  in  opposite  mem- 
bers of  an  equation. 

Alkhwarizmfs  classification  of  equations  of  the  1st  and 
2d  degrees  is  that  to  which  these  processes  would  naturally 
lead,  viz. : 

ax2  =  bx,         bx2  =  c,  bx  =  c, 

x2  +  bx  —  c,     x2  -f  c  =  bx,     x2=bx-\-  c. 

These  equations  he  solves  separately,  following  up  the 
solution  in  each  case  with  a  geometric  demonstration  of  its 


ARABIAN  ALGEBRA.  109 

correctness.  He  recognizes  both  roots  of  the  quadratic 
when  they  are  positive.  In  this  respect  he  is  Indian ;  in 
all  others — the  avoidance  of  negatives,  the  nse  of  geometric 
demonstration  —  he  is  Greek. 

Besides  Alkhwarizml,  the  most  famous  algebraists  of  the 
Arabians  were   Alkarchi   and  Alchayydmt,  both   of  whom 
lived  in  the  11th  century. 
r    Alkarchi  gave  the  solution  of  equations  of  the  forms : 

ax2p  +  bxp  =  c,  ax2p  +  c  =  bxp,  bxp  +  c  =  ax2p. 

■\     He  also  reckoned  with  irrationals,  the  equations 

V8  +  Vl8  =  V50,  </M-</2=Vl6, 

being  pretty  just  illustrations  of  his  success  in  this  field. 
, — - — sAlchayyami  was  the  first  mathematician  to  make  a  system- 
atic investigation  of  the  cubic  equation.  He  classified  the 
various  forms  which  this  equation  takes  when  all  its  terms 
are  positive,  and  solved  each  form  geometrically  —  by  the 
intersections  of  conies. #  A  pure  algebraic  solution  of  the 
cubic  he  believed  impossible. 

Like  Alkhwarizmi,  Alkarchi  and  Alchayyami  were  Eastern 
Arabians.  But  early  in  the  8th  century  the  Arabians  con- 
quered a  great  part  of  Spain.  An  Arabian  realm  was 
established  there  which  became  independent  of  the  Bagdad 
caliphate  in  747,  and  endured  for  300  years.  The  inter- 
course of  these  Western  Arabians  with  the  East  was  not 


*  Thus  suppose  the  equation  xs  +  bx  ==  a,  given. 

For  b  substitute  the  quantity  />2,  and  for  o,  p2r.    Then  xs  =  p2(r  —  x). 

Now  this  equation  is  the  result  of  eliminating  y  from  between  the 
two  equations,  x2  =  py,  y2  =  x  (r  —  x)  ;  the  first  of  which  is  the  equa- 
tion of  a  parabola,  the  second,  of  a  circle. 

Let  these  two  curves  be  constructed  ;  they  will  intersect  in  one  real 
point  distinct  from  the  origin,  and  the  abscissa  of  this  point  is  a  root 
of  xz  -f-  bx  =  a.     See  Hankel,  Geschichte  der  Mathematik,  p.  279. 

This  method  is  of  greater  interest  in  the  history  of  geometry  than 
in  that  of  algebra.  It  involves  an  anticipation  of  some  of  the  most 
important  ideas  of  Descartes'  Geometric  (see  p.  118). 


110  NUMBER-SYSTEM  OF  ALGEBRA. 

frequent  enough  to  exercise  a  controlling  influence  on  their 
aesthetic  or  scientific  development.  Their  mathematical 
productions  are  of  a  later  date  than  those  of  the  East 
and  almost  exclusively  arithmetico-algebraic.  They  con- 
structed a  formal  algebraic  notation  which  went  over  into 
the  Latin  translations  of  their  writings  and  rendered  the 
path  of  the  Europeans  to  a  knowledge  of  the  doctrine  of 
equations  easier  than  it  would  have  been,  had  the  Arabians 
of  the  East  been  their  only  instructors.  The  best  known  of 
their  mathematicians  are  Ibn  Aflah  (end  of  11th  century), 
Ibn  Albanna  (end  of  13th  century),  Alkasadl  (15th  century). 

109.  Arabian  Algebra  Greek  rather  than  Indian.  Thus, 
of  the  three  greater  departments  of  the  Arabian  mathematics, 
the  Indian  influence  gained  the  mastery  in  reckoning^only. 

The  Arabian  geometry  is  Greek  through  and  through. 

While  the  algebra  contains  both  elements,  the  Greek  pre- 
dominates. Indeed,  except  that  both  roots  of  the  quadratic 
are  recognized,  the  doctrine  of  the  determinate  equation  is 
altogether  Greek.  It  avoids  the  negative  almost  as  care- 
fully as  Diophantus  does ;  and  in  its  use  of  the  geometric 
method  of  demonstration  it  is  actuated  by  a  spirit  less 
modern  still  —  the  spirit  in  which  Euclid  may  have  con- 
ceived of  algebra  when  he  solved  his  geometric  quadratics. 

The  theory  of  indeterminate  equations  seldom  goes  beyond 
Diophantus ;  where  it  does,  it  is  Indian. 

The  Arabian  trigonometry  is  based  on  Ptolemy's,  but  is 
its  superior  in  two  important  particulars.  It  employs  the 
sine  where  Ptolemy  employs  the  chord  (being  in  this  re- 
spect Indian),  and  has  an  algebraic  instead  of  a  geometric 
form.  Some  of  the  methods  of  approximation  used  in 
reckoning  out  trigonometric  tables  show  great  cleverness. 
Indeed,  the  Arabians  make  some  amends  for  their  ill-advised 
return  to  geometric  algebra  by  this  excellent  achievement 
in  algebraic  geometry. 

The  preference  of  the  Arabians  for  Greek  algebra  was 


EARLY  EUROPEAN  ALGEBRA.  Ill 

especially  unfortunate  in  respect  to  the  negative,  which  was 
in  consequence  forced  to  repeat  in  Europe  the  fight  for 
recognition  which  it  had  already  won  in  India. 

110.  Mathematics  in  Europe  before  the  Twelfth  Century. 

The  Arabian  mathematics  found  entrance  to  Christian  Eu- 
rope in  the  12th  century.  During  this  century  and  the  first 
half  of  the  next  a  good  part  of  its  literature  was  translated 
into  Latin. 

Till  then  the  plight  of  mathematics  in  Europe  had  been 
miserable  enough.  She  had  no  better  representatives  than 
the  Romans,  the  most  deficient  in  the  sense  for  mathematics 
of  all  cultured  peoples,  ancient  or  modern ;  no  better  lit- 
erature than  the  collection  of  writings  on  surveying  known 
as  the  Codex  Arcerianus,  and  the  childish  arithmetic  and 
geometry  of  Boetius. 

Prior  to  the  10th  century,  however,  Northern  Europe  had 
not  sufficiently  emerged  from  barbarism  to  call  even  this 
paltry  mathematics  into  requisition.  What  learning  there 
was  was  confined  to  the  cloisters.  Beckoning  (computus) 
was  needed  for  the  Church  calendar  and  was  taught  in  the 
cloister  schools  established  by  Alcuin  (735-804)  under  the 
patronage  of  Charlemagne.  Reckoning  was  commonly  done 
on  the  fingers.  Not  even  was  the  multiplication  table  gen- 
erally learned.  Reference  would  be  made  to  a  written  copy 
of  it,  as  nowadays  reference  is  made  to  a  table  of  loga- 
rithms. The  Church  did  not  need  geometry,  and  geometry 
in  any  proper  sense  did  not  exist. 

111.  Gerbert.  But  in  the  10th  century  there  lived  a  man 
of  true  scientific  interests  and  gifts,  Gerbert,*  Bishop  of 
Rheims,  Archbishop  of  Ravenna,  and  finally  Pope  Sylvester 
II.  In  him  are  the  first  signs  of  a  new  life  for  mathematics. 
His  achievements,  it  is  true,  do  not  extend  beyond  the 
revival  of  Roman  mathematics,  the  authorship  of  a  geom- 

*  See  §  88. 


112  NUMBEBSYSTEM  OF  ALGEBBA. 

etry  based  on  the  Codex  Arcerianus,  and  a  method  for  effect- 
ing division  on  the  abacus  with  apices.  Yet  these  achieve- 
ments are  enough  to  place  him  far  above  his  contemporaries. 
His  influence  gave  a  strong  impulse  to  mathematical  studies 
where  interest  in  them  had  long  been  dead.  He  is  the  fore- 
runner of  the  intellectual  activity  ushered  in  by  the  trans- 
lations from  the  Arabic,  for  he  brought  to  life  the  feeling 
of  the  need  for  mathematics  which  these  translations  were 
made  to  satisfy. 

112.   Entrance  of  the  Arabian  Mathematics.     Leonardo. 

It  was  the  elementary  branch  of  the  Arabian  mathematics 
which  took  root  quickest  in  Christendom  —  reckoning  with 
9  digits  and  0. 

Leonardo  of  Pisa  —  Fibonacci,  as  he  was  also  called  — 
did  great  service  in  the  diffusion  of  the  new  learning 
through  his  Liber  Abaci  (1202  and  1228),  a  remarkable 
presentation  of  the  arithmetic  and  algebra  of  the  Arabians, 
which  remained  for  centuries  the  fund  from  which  reckoners 
and  algebraists  drew  and  is  indeed  the  foundation  of  the 
modern  science. 

The  four  fundamental  operations  on  integers  and  frac- 
tions are  taught  after  the  Arabian  method ;  the  extraction  of 
the  square  root  and  the  doctrine  of  irrationals  are  presented 
in  their  pure  algebraic  form ;  quadratic  equations  are  solved 
and  applied  to  quite  complicated"  problems ;  negatives  are 
accepted  when  they  admit  of  interpretation  as  debt. 

The  last  fact  illustrates  excellently  the  character  of  the 
Liber  Abaci.  It  is  not  a  mere  translation,  but  an  inde- 
pendent and  masterly  treatise  in  one  department  of  the 
new  mathematics. 

Besides  the  Liber  Abaci,  Leonardo  wrote  the  Practica 
Geometriae,  which  contains  much  that  is  best  of  Euclid, 
Archimedes,  Hero,  and  the  elements  of  trigonometry ;  also 
the  Liber  Quadratorum,  a  collection  of  original  algebraic 
problems  most  skilfully  handled. 


EARLY  EUROPEAN  ALGEBRA.  113 

113.  Mathematics  during  the  Age  of  Scholasticism.  Leo- 
nardo was  a  great  mathematician,*  but  tine  as  his  work  was, 
it  bore  no  fruit  until  the  end  of  the  15th  century.  In  him 
there  had  been  a  brilliant  response  to  the  Arabian  impulse. 
But  the  awakening  was  only  momentary  ;  it  quickly  yielded 
to  the  heavy  lethargy  of  the  "  dark  "  ages. 

The  age  of  scholasticism,  the  age  of  devotion  to  the  forms 
of  thought,  logic  and  dialectics,  is  the  age  of  greatest  dul- 
ness  and  confusion  in  mathematical  thinking.!  Algebra 
owes  the  entire  period  but  a  single  contribution;  the 
concept  of  the  fractional  power.  Its  author  was  Nicole 
Oresme  (died  1382),  who  also  gave  a  symbol  for  it  and  the 
rules  by  which  reckoning  with  it  are  governed. 

*  Besides  Leonardo  there  flourished  in  the  first  quarter  of  the  13th 
century  an  able  German  mathematician,  Jordanus  Nemorarius.  He 
was  the  author  of  a  treatise  entitled  Be  numeris  datis,  in  which  known 
quantities  are  for  the  first  time  represented  by  letters,  and  of  one  De 
trangulis  which  is  a  rich  though  rather  systemless  collection  of  theorems 
and  problems  principally  of  Greek  and  Arabian  origin.  See  Giinther : 
Geschichte  des  mathemathischen  Unterrichts  im  deutschen  Mittelalter, 
p.  156. 

t  Compare  Hankel,  Geschichte  der  Mathematik,  pp.  349-352.  To 
the  unfruitfulness  of  these  centuries  the  Summa  of  Luca  Pacioli  bears 
witness.  This  book^  which  has  the  distinction  of  being  the  earliest 
book  on  algebra  printed,  appeared  in  1494,  and  embodies  the  arith- 
metic, algebra,  and  geometry  of  the  time  just  preceding  the  Renais- 
sance. It  contains  not  an  idea  or  method  not  already  presented  by 
Leonardo.  Even  in  respect  to  algebraic  symbolism  it  surpasses  the 
Liber  Abaci  only  to  the  extent  of  using  abbreviations  for  a  few  fre- 
quently recurring  words,  as  p.  for  "plus,"  and  R.  for  "res"  (the 
unknown  quantity) .  And  this  is  not  to  be  regarded  as  original  with 
Pacioli  for  the  Arabians  of  Leonardo's  time  made  a  similar  use  of 
abbreviations.  In  a  translation  made  by  Gerhard  of  Cremona  (12th 
century)  from  an  unknown  Arabic  original  the  letters  r  (radix), 
c  (census) ,  d  (dragma)  are  used  to  represent  the  unknown  quantity, 
its  square,  and  the  absolute  term  respectively. 

Pacioli' s  demonstration  that  "minus  times  minus  is  plus"  is  per- 
haps worth  inserting  here,  not,  unfortunately,  because  it  has  gone 
altogether  out  of  vogue,  but  for  the  sake  of  the  scholastic  principle  on 


114  NUMBER-SYSTEM  OF  ALGEBRA. 

114.  The  Renaissance.  Solution  of  the  Cubic  and  Bi- 
quadratic Equations.  The  first  achievement  in  algebra  by 
the  mathematicians  of  the  Renaissance  was  the  algebraic 
solntion  of  the  cubic  equation :  a  fine  beginning  of  a  new 
era  in  the  history  of  the  science. 

The  cubic  xs  -f-  «W3  =  n  was  solved  by  Ferro  of  Bologna 
in  1505,  and  a  second  time  and  independently,  in  1535,  by 
Ferro's  countryman,  Tartaglia,  who  by  help  of  a  transfor- 
mation made  his  method  apply  to  xs  ±  mx2  =  ±  n  also. 
But  Cardan  of  Milan  was  the  first  to  publish  the  solution, 
in  his  Ars  Magna,*  1545. 

The  Ars  Magna  records  another  brilliant  discovery :  the 
solution  —  after  a  general  method  —  of  the  biquadratic 
x4  +  6x2  +  36  =  60 x  by  Ferrari,  a  pupil  of  Cardan. 

Thus  in  Italy,  within  fifty  years  of  the  new  birth  of 
algebra,  after  a  pause  of  sixteen  centuries  at  the  quad- 
ratic, the  limits  of  possible  attainment  in  the  algebraic 
solution  of  equations  were  reached;  for  the  algebraic 
solution  of  the  general  equation  of  a  degree  higher  than 
4  is  impossible,  as  was  first  demonstrated  by  Abel.f 

The  general  solution  of  higher  equations  proving  an 
obstinate  problem,  nothing  was  left  the  searchers  for  the 

which  he  bases  it.  He  reasons  thus  :  Since  8  •  8  =  (10  —  2)  (10  —  2)  =  64, 
and  10  •  10  =  100,  and  -  2  .  10  =  -  20  ;  therefore,  -  2  •  -  2  =  +  4  —  and 
adds  that  this  method  of  reasoning  is  well-known  to  philosophers, 
being  "  a  disjunctiva  plurium  partium  a  destructione  multarum  supra 
unam  semper  tenet  consequentia. " 

It  should  be  added  that  the  loth  century  produced  a  mathemati- 
cian who  deserves  a  distinguished  place  in  the  general  history  of 
mathematics  on  account  of  his  contributions  to  trigonometry,  the 
astronomer  Regiomontanus  (1436-1476).  Like  Jordanus,  he  was  a 
German. 

*  The  proper  title  of  this  work  is  :  "  Artis  magnae  sive  de  regulis 
Algebraicis  liber  unus."  It  has  stolen  the  title  of  Cardan's  "Ars 
magna  Arithmeticae,"  published  at  Basel,  1570. 

t  Memoire  sur  les  Equations  Algebriques  :  Christiania,  1826.  Also 
in  Crelle's  Journal,  I,  p.  65. 


EABLY  EUROPEAN  ALGEBEA.  115 

roots  of  equations  but  to  devise  a  method  of  working  them 
out  approximately.  In  this  the  French  mathematician 
Vieta  (1540-1603)  was  successful,  his  method  being  essen- 
tially the  same  as  that  now  known  as  Newton's. 

115.  The  Negative  in  the  Algebra  of  this  Period.  First 
Appearance  of  the  Imaginary.  But  the  general  equation 
presented  other  problems  than  the  discovery  of  rules  for 
obtaining  •  its  roots;  the  nature  of  these  roots  and  the 
relations  between  them  and  the  coefficients  of  the  equation 
invited  inquiry. 

We  witness  another  phase  of  the  struggle  of  the  negative 
for  recognition.  The  imaginary  is  now  ready  to  make  com- 
mon cause  with  it. 

Already  in  the  Ars  Magna  Cardan  distinguishes  between 
numeri  veri — the  positive  integer,  fraction,  and  irrational, — 
and  numeri  ficti,  or  falsi  —  the  negative  and  the  square  root 
of  the  negative.  Like  Leonardo,  he  tolerates  negative  roots 
of  equations  when  they  admit  of  interpretation  as  "  debitum," 
not  otherwise.  While  he  has  no  thought  of  accepting  im- 
aginary roots,  he  shows  that  if  5  + V— 15  be  substituted 
for  x  in  x  (10  —  x)  =  40,  that  equation  is  satisfied ;  which, 
of  course,  is  all  that  is  meant  nowadays  when  5  4-  V—  15 
is  called  a  root.  His  declaration  that  5  ±  V— 15  are 
"  vere  sophistica  "  does  not  detract  from  the  significance  of 
this,  the  earliest  recorded  instance  of  reckoning  with  the 
imaginary.  It  ought  perhaps  to  be  added  that  Cardan  is 
not  always  so  successful  in  these  reckonings ;  for  in  another 
place  he  sets 


4\      \      AJ      \64      8* 


4\ 

Following  Cardan,  Bombelli*  reckoned  with  imaginaries 
to  good  purpose,  explaining  by  their  aid  the  irreducible 
case  in  Cardan's  solution  of  the  cubic. 

*  L' Algebra,  1579.  He  also  formally  states  rules  for  reckoning 
with  ±  V  —  1  and  a  +  b  V  —  1. 


116  NUMBER-SYSTEM  OF  ALGEBRA. 

On  the  other  hand,  neither  Yieta  nor  his  distinguished 
follower,  the  Englishman  Harriot  (1560-1621),  accept  even 
negative  roots ;  though  Harriot  does  not  hesitate  to  perform 
algebraic  reckonings  on  negatives,  and  even  allows  a  nega- 
tive to  constitute  one  member  of  an  equation. 

116.  Algebraic  Symbolism.  Vieta  and  Harriot.  Yieta 
and  Harriot,  however,  did  distinguished  service  in  perfect- 
ing the  symbolism  of  algebra ;  Yieta,  by  the  systematic  use 
of  letters  to  represent  known  quantities,  —  algebra  first 
became  "literal"  or  "universal  arithmetic"  in  his  hands, * 
—  Harriot,  by  ridding  algebraic  statements  of  every  non- 
symbolic  element,  of  everything  but  the  letters  which  rep- 
resent quantities  known  as  well  as  unknown,  symbols  of 
operation,  and  symbols  of  relation.  Harriot's  Artis  Analy- 
ticae  Praxis  (1631)  has  quite  the  appearance  of  a  modern 
algebra,  f 

*  There  are  isolated  instances  of  this  use  of  letters  much  earlier  than 
Vieta  in  the  De  numeris  datis  of  Jordanus  Nemorarius,  and  in  the 
AJgorithnus  demonstrates  of  Regiomontanus.  But  the  credit  of  making 
it  the  general  practice  of  algebraists  belongs  to  Vieta. 

t  One  has  only  to  reflect  how  much  of  the  power  of  algebra  is  due 
to  its  admirable  symbolism  to  appreciate  the  importance  of  the  Artis 
Analyticae  Praxis,  in  which  this  symbolism  is  finally  established.  But 
one  addition  of  consequence  has  since  been  made  to  it,  integral  and 
fractional  exponents  introduced  by  Descartes  (1637)  and  Wallis  (1659). 

Harriot  substituted  small  letters  for  the  capitals  used  by  Vieta,  but 
followed  Vieta  in  representing  known  quantities  by  consonants  and 
unknown  by  vowels.  The  present  convention  of  representing  known 
quantities  by  the  earlier  letters  of  the  alphabet,  unknown  by  the  later, 
is  due  to  Descartes. 

Vieta' s  notation  is  unwieldy  and  ill  adapted  to  purposes  of  alge- 
braic reckoning.  Instead  of  restricting  itself,  as  Harriot's  does,  to 
the  use  of  brief  and  easily  apprehended  conventional  symbols,  it  also 
employs  words  subject  to  the  rules  of  syntax.  Thus  for  As  —  3  B1  A  =  Z 
(or  aaa  —  3  bba  =  z,  as  Harriot  would  have  written  it) ,  Vieta  writes 
A  cubus  —  B  quad  3  in  A  aeqnatur  Z  solido.  In  this  respect  Vieta  is 
inferior  not  only  to  Harriot,  but  to  several  of  his  predecessors  and 


EARLY  EUROPEAN  ALGEBRA.  117 

117.  Fundamental  Theorem  of  Algebra.  Harriot  and 
Girard.  Harriot  has  been  credited  with  the  discovery  of 
the  "fundamental  theorem"  of  algebra  — the  theorem  that 
the  number  of  roots  of  an  algebraic  equation  is  the  same  as 
its  degree.  The  Artis  Analyticae  Praxis  contains  no  mention 
of  this  theorem  —  indeed,  by  ignoring  negative  and  imagi- 
nary roots,  leaves  no  place  for  it;  yet  Harriot  develops 
systematically  a  method  which,  if  carried  far  enough,  leads 
to  the  discovery  of  this  theorem  as  well  as  to  the  relations 
holding  between  the  roots  of  an  equation  and  its  coefficients. 

By  multiplying  together  binomial  factors  which  involve 
the  unknown  quantity,  and  setting  their  product  equal  to 
0,  he  builds  "canonical"  equations,  and  shows  that  the 
roots  of  these  equations  —  the  only  roots,  he  says  —  are  the 
positive  values  of  the  unknown  quantity  which  render  these 
binomial  factors  0.  Thus  he  builds  act  —  ba  —  ca=  —  be, 
in  which  a  is  the  unknown  quantity,  out  of  the  factors 
a  —  b,  a  +  c,  and  proves  that  &  is  a  root  of  this  equation  and 
the  only  root,  the  negative  root  c  being  totally  ignored. 

While  no  attempt  is  made  to  show  that  if  the  terms  of 
a  "common"  equation  be  collected  in  one  member,  this  can 

notably  to  his  contemporary,  the  Dutch  mathematician  Stevinus 
(1548-1620),  who  would,  for  instance,  have  written  x2  +  3x  —  8  as 
l®  +  3®  —  8®.  The  geometric  affiliations  of  Vieta's  notation  are 
obvious.     It  suggests  the  Greek  arithmetic. 

It  is  surprising  that  algebraic  symbolism  should  owe  so  little  to  the 
great  Italian  algebraists  of  the  16th  century.  Like  Pacioli  (see  note, 
p.  113)  they  were  content  with  a  few  abbreviations  for  words,  a 
"syncopated"  notation,  as  it  has  been  called,  and  an  incomplete  one 
at  that. 

The  current  symbols  of  operation  and  relation  are  chiefly  of  English 
and  German  origin,  having  been  invented  or  adopted  as  follows :  viz. 
=,  by  Recorde  in  1540  ;  ^/,  by  Rudolf  in  1526  ;  the  vinculum,  by  Vieta 
in  1591  ;  brackets,  by  Girard  in  1629  ;  -f-,  by  Pell  in  1630  ;  X,  >,  <,  by 
Harriot  in  1631.  The  signs  -f  and  —  occur  in  a  15th  century  manu- 
script discovered  by  Gerhardt  at  Vienna.  The  notations  a  —  b  and  - 
for  the  fraction  were  adopted  from  the  Arabians. 


118  NUMBER-SYSTEM  OF  ALGEBRA. 

be  separated  into  binomial  factors,  the  case  of  canonical 
equations  raised  a  strong  presumption  for  the  soundness  of 
this  view  of  the  structure  of  an  equation. 

The  first  statement  of  the  fundamental  theorem  and 
of  the  relations  between  coefficients  and  roots  occurs  in 
a  remarkably  clever  and  modern  little  book,  the  Inven- 
tion Nouvelle  en  VAlyebre,  of  Albert  Girard,  published  in 
Amsterdam  in  1629,  two  years  earlier,  therefore,  than  the 
Artis  Analyticae  Praxis.  Girard  stands  in  no  fear  of  imag- 
inary roots,  but  rather  insists  on  the  wisdom  of  recognizing 
them.  They  never  occur,  he  says,  except  when  real  roots 
are  lacking,  and  then  in  number  just  sufficient  to  fill  out 
the  entire  number  of  roots  to  equality  with  the  degree  of 
the  equation. 

Girard  also  anticipated  Descartes  in  the  geometrical  in- 
terpretation of  negatives.  But  the  Invention  Nouvelle  does 
not  seem  to  have  attracted  much  notice,  and  the  genius  and 
authority  of  Descartes  were  needed  to  give  the  interpreta- 
tion general  currency. 


VI.    ACCEPTANCE    OF   THE  NEGATIVE,   THE    GENEKAL 
IEKATIONAL,  AND  THE  IMAGINAKY  AS  NUMBEKS. 

118.  Descartes'  Geometrie  and  the  Negative.  The  Geome- 
trie of  Descartes  appeared  in  1637.  This  famous  little  trea- 
tise enriched  geometry  with  a  general  and  at  the  same  time 
simple  and  natural  method  of  investigation :  the  method  of 
representing  a  geometric  curve  by  an  equation,  which,  as 
Descartes  puts  it,  expresses  generally  the  relation  of  its 
points  to  those  of  some  chosen  line  of  reference.*  To  form 
such  equations  Descartes  represents  line  segments  by  letters, 
—  the  known  by  a,  b,  c,  etc.,  the  unknown  by  x  and  y.     He 

*  See  Geometrie,  Livre  II.  In  Cousin's  edition  of  Descartes'  works, 
Vol.  V,  p.  337. 


THE  NEGATIVE.  119 

supposes  a  perpendicular,  y,  to  be  dropped  from  any  point 
of  the  curve  to  the  line  of  reference,  and  then  the  equation 
to  be  found  from  the  known  properties  of  the  curve  which 
connects  y  with  x,  the  distance  of  y  from  a  fixed  point  of  the 
line  of  reference.  This  is  the  equation  of  the  curve  in  that 
it  is  satisfied  by  the  x  and  y  of  each  and  every  curve-point. # 
To  meet  the  difficulty  that  the  mere  length  of  the  perpen- 
dicular (y)  from  a  curve-point  will  not  indicate  to  which 
side  of  the  line  of  reference  the  point  lies,  Descartes  makes 
the  convention  that  perpendiculars  on  opposite  sides  of  this 
line  (and  similarly  intercepts  (x)  on  opposite  sides  of  the 
point  of  reference)  shall  have  opposite  algebraic  signs. 

This  convention  gave  the  negative  a  new  position  in 
mathematics.  Not  only  was  a  "real"  interpretation  here 
found  for  it,  the  lack  of  which  had  made  its  position  so  dif- 
ficult hitherto,  but  it  was  made  indispensable,  placed  on  a 
footing  of  equality  with  the  positive.  The  acceptance  of 
the  negative  in  algebra  kept  pace  with  the  spread  of  Descar- 
tes' analytical  method  in  geometry. 

119.  Descartes'  Geometric  Algebra.  But  the  Geometrie 
has  another  and  perhaps  more  important  claim  on  the  atten- 
tion of  the  historian  of  algebra.  The  entire  method  of  the 
book  rests  on  the  assumption  —  made  only  tacitly,  to  be 
sure,  and  without  knowledge  of  its  significance  —  that  two 
algebras  are  formally  identical  whose  fundamental  opera- 
tions are  formally  the  same ;  i.e.  subject  to  the  same  laws 
of  combination. 

For  the  algebra  of  the  Geome'trie  is  not,  as  is  commonly 
said,  mere  numerical  algebra,  but  what  may  for  want  of  a 
better  name  be  called  the  algebra  of  line  segments.  Its 
symbolism  is  the  same  as  that  of  numerical  algebra;  but 

*  Descartes  fails  to  recognize  a  number  of  the  conventions  of  our 
modern  Cartesian  geometry.  He  makes  no  formal  choice  of  two  axes 
of  reference,  calls  abscissas  y  and  ordinates  x,  and  as  frequently  regards 
as  positive  ordinates  below  the  axis  of  abscissas  as  ordinates  above  it. 


120  NUMBER-SYSTEM  OF  ALGEBRA. 

symbols  which  there  represent  numbers  here  represent  line 
segments.  Not  only  is  this  the  case  with  the  letters  a,  b,  x, 
y,  etc.,  which  are  mere  names  (noms)  of  line  segments,  not 
their  numerical  measures,  but  with  the  algebraic  combina- 
tions of  these  letters,  a  +  b  and  a  —  b  are  respectively 
the  sum  and  difference  of  the  line  segments  a  and  b;  ab,  the 

fourth  proportional  to  an  assumed  unit  line,  a,  and  b ;  -,  the 

b 
fourth  proportional  to  b,  a,  and  the  unit  line ;  and  Va,  Va, 
etc.,  the  first,  second,  etc.,  mean  proportionals  to  the  unit 
line  and  a.* 

Descartes'  justification  of  this  use  of  the  symbols  of 
numerical  algebra  is  that  the  geometric  constructions  of 
which  he  makes  a  +  b,  a  —  b,  etc.,  represent  the  results  are 
"  the  same  "  as  numerical  addition,  subtraction,  multiplica- 
tion, division,  and  evolution,  respectively.  Moreover,  since 
all  geometric  constructions  which  determine  line  segments 
may  be  resolved  into  combinations  of  these  constructions 
as  the  operations  of  numerical  algebra  into  the  fundamental 
operations,  the  correspondence  which  holds  between  these 
fundamental  constructions  and  operations  holds  equally 
between  the  more  complex  constructions  and  operations. 
The  entire  system  of  the  geometric  constructions  under 
consideration  may  therefore  be  regarded  as  formally  iden- 
tical with  the  system  of  algebraic  operations,  and  be 
represented   by  the  same  symbolism. 

In  what  sense  his  fundamental  constructions  are  "the 
same "  as  the  fundamental  operations  of  arithmetic,  Des- 
cartes does  not  explain.  The  true  reason  of  their  formal 
identity  is  that  both  are  controlled  by  the  commutative, 
associative,  and  distributive  laws.  Thus  in  the  case  of  the 
former  as  of  the  latter,  ab  =  ba,  and  a  (be)  =abc,  for  the 
fourth  proportional  to  the  unit  line,  a,  and  b  is  the  same  as 
the  fourth  proportional  to  the  unit  line,  b,  and  a ;  and  the 
fourth  proportional  to  the  unit  line,  a,  and  be  is  the  same  as 

*  Geometrie,  Livre  I.     Ibid.  pp.  313-314. 


THE  GENERAL  IRRATIONAL.  121 

the  fourth  proportional  to  the  unit  line,  ab,  and  c.  But  this 
reason  was  not  within  the  reach  of  Descartes,  in  whose  day 
the  fundamental  laws  of  numerical  algebra  had  not  yet 
been  discovered. 

120.  The  Continuous  Variable.     Newton.    Euler.     It  is 

customary  to  credit  the  Geometrie  with  having  introduced 
the  continuous  variable  into  mathematics,  but  without  suffi- 
cient reason.  Descartes  prepared  the  way  for  this  con- 
cept, but  he  makes  no  use  of  it  in  the  Geometrie.  The  x 
and  y  which  enter  in  the  equation  of  a  curve  he  regards  not 
as  variables  but  as  indeterminate  constants,  a  pair  of  whose 
values  correspond  to  each  curve-point.*  The  real  author  of 
this  concept  is  Newton  (1642-1727),  of  whose  great  inven- 
tion, the  method  of  fluxions,  continuous  variation,  "flow," 
is  the  fundamental  idea. 

But  Newton's  calculus,  like  Descartes'  algebra,  is  geo- 
metric rather  than  purely  numerical,  and  his  followers  in 
England,  as  also,  to  a  less  extent,  the  followers  of  his  great 
rival,  Leibnitz,  on  the  continent,  in  employing  the  calculus, 
for  the  most  part  conceive  of  variables  as  lines,  not  num- 
bers. The  geometric  form  again  threatened  to  become  para- 
mount in  mathematics,  and  geometry  to  enchain  the  new 
"  analysis  "  as  it  had  formerly  enchained  the  Greek  arith- 
metic. It  is  the  great  service  of  Euler  (1707-1783)  to  have 
broken  these  fetters  once  for  all,  to  have  accepted  the  con- 
tinuously variable  number  in  its  purity,  and  therewith  to 
have  created  the  pure  analysis.  For  the  relations  of  con- 
tinuously variable  numbers  constitute  the  field  of  the  pure 
analysis  ;  its  central  concept,  the  function,  being  but  a  device 
for  representing  their  interdependence. 

121.  The  General  Irrational.  While  its  concern  with 
variables  puts  analysis  in  a  certain  opposition  to  elementary 
algebra,  concerned  as  this  is  with  constants,  its  establish- 

*  Geometrie,  Livre  II.     Ibid.  pp.  337-338. 


122  NUMBEB-SYSTEM  OF  ALGEBRA. 

ment  of  the  continuously  variable  number  in  mathematics 
brought  about  a  rich  addition  to  the  number-system  of  alge- 
bra—  the  general  irrational.  Hitherto  the  only  irrational 
numbers  had  been  "surds,"  impossible  roots  of  rational 
numbers ;  henceforth  their  domain  is  as  wide  as  that  of  all 
possible  lines  incommensurable  with  any  assumed  unit  line. 

122.  The  Imaginary,  a  Recognized  Analytical  Instrument. 

Out  of  the  excellent  results  of  the  use  of  the  negative  grew 
a  spirit  of  toleration  for  the  imaginary.  Increased  atten- 
tion was  paid  to  its  properties.  Leibnitz,  noticed  the  real 
sum  of  conjugate  imaginaries  (1676-7)  ;  Demoivre  dis- 
covered (1730)  the  famous  theorem 

(cos  6  -\-  i  sin  0)n  =  cos  nO  +  i  sin  nO ; 

and  Euler  (1748)  the  equation 

cos  &  +  i  sin  0  ==  e<9, 

which  plays  so  great  a  r61e  in  the  modern  theory  of 
functions. 

Euler  also,  practising  the  method  of  expressing  complex 
numbers  in  terms  of  modulus  and  angle,  formed  their  prod- 
ucts, quotients,  powers,  roots,  and  logarithms,  and  by  many 
brilliant  discoveries  multiplied  proofs  of  the  power  of  the 
imaginary  as  an  analytical  instrument. 

123.  Argand's  Geometric  Representation  of  the  Imaginary. 

But  the  imaginary  was  never  regarded  as  anything  better 
than  an  algebraic  fiction  —  to  be  avoided,  where  possible, 
by  the  mathematician  who  prized  purity  of  method — until 
the  discovery  of  a  geometric  picture  for  it  such  as  that  with 
which  Descartes  had  supplied  the  negative.  The  first  to 
render  it  this  service  was  a  French  mathematician,  Argand, 
in  1806.* 

*  Essai  sur  une  maniere  de  representer  les  quantit€s  imaginaires  dans  les 
constructions  geometriques. 


THE  COMPLEX  NUMBER.  123 

As  -f- 1  and  —  1  may  be  represented  by  unit  lines  drawn 
in  opposite  directions  from  any  point,  0,  and  as  t  (i.e.  V  — 1) 
is  a  mean  proportional  to  + 1  and  —  1,  it  occurred  to  Argand 
to  represent  this  symbol  by  the  line  whose  direction  with 
respect  to  the  line  + 1  is  the  same  as  the  direction  of  the 
line  —1  with  respect  to  it;  viz.,  the  unit  perpendicular 
through  0  to  the  1  —  line.  Let  only  the  direction  of  the 
1  —  line  be  fixed,  the  position  of  the  point  0  in  the  plane 
is  altogether  indifferent. 

Between  the  segments  of  a  given  line,  whether  taken  in 
the  same  or  opposite  directions,  the  equation  holds ; 

AB  +  BC=AC 

It  means  nothing  more,  however,  when  the  directions  of 
AB  and  BG  are  opposite,  than  that  the  result  of  carrying 
a  moving  point  from  A  first  to  B,  and  thence  back  to  <7,  is 
the  same  as  carrying  it  from  A  direct  to  C.  But  in  this 
sense  the  equation  holds  equally  when  A,  B,  C  are  not  in 
the  same  right  line. 

Given,  therefore,  a  complex  number,  a  +  ib ;  choose  any 
point  A  in  the  plane ;  from  it  draw  a  line  AB,  of  length  a, 
in  the  direction  of  the  1  —  line,  and  from  B  a  line  BO,  of 
length  b,  in  the  direction  of  the  %  —  line.  The  line  AC,  thus 
fixed  in  length  and  direction,  but  situated  anywhere  in  the 
plane,  is  Argand's  picture  of  a  -f-  ib.    ■ 

Argand's  skill  in  the  use  of  his  new  device  was  equal  to 
the  discovery  of  the  demonstration  given  in  §  54,  that  every 
algebraic  equation  has  a  root. 

124.  Gauss.  The  Complex  Number.  The  method  of  rep- 
resenting complex  numbers  in  common  use  to-day,  that 
described  in  §  42,  is  due  to  Gauss.  He  was  already  in  pos- 
session of  it  in  1811,  though  he  published  no  account  of  it 
until  1831. 

To  Gauss  belongs  the  conception  of  i  as  an  independent 
unit  co-ordinate  with  1,  and  of  a  -f-  ib  as  a  complex  number, 


/ 

124  NUMBER-SYSTEM  OF  ALGEBRA. 

a  sum  of  multiples  of  the  units  1  and  %\  his  also  is  the 
name  "  complex  number  "  and  the  concept  of  complex  num- 
bers in  general,  whereby  a  +  ib  secures  a  footing  in  the 
theory  of  numbers  as  well  as  in  algebra. 

He  too,  and  not  Argand,  must  be  credited  with  really 
breaking  down  the  opposition  of  mathematicians  to  the 
imaginary.  Argand' s  Essai  was  little  noticed  when  it  ap- 
peared, and  soon  forgotten ;  but  there  was  no  withstanding 
the  great  authority  of  Gauss,  and  his  precise  and  masterly 
presentation  of  this  doctrine.* 


VII.     EECOGNITION  OP  THE   PUKELY  SYMBOLIC 
CHAEACTEE  OF  ALGEBEA. 

QUATERNIONS.    AUSDEHNUNGSLEHRE. 

125.  The  Principle  of  Permanence.  Thus,  one  after 
another,  the  fraction,  irrational,  negative,  and  imaginary, 
gained  entrance  to  the  number-system  of  algebra.  iNot 
one  of  them  was  accepted  until  its  correspondence  to  some 
actually  existing  thing  had  been  shown,  the  fraction  and 
irrational,  which  originated  in  relations  among  actually  ex- 
isting things,  naturally  making  good  their  position  earlier 
than  the  negative  and  imaginary,  which  grew  immediately 
out  of  the  equation,  and  for  which  a  "  real "  interpretation 
had  to  be  sought. 

Inasmuch  as  this  correspondence  of  the  artificial  numbers 
to  things  extra-arithmetical,  though  most  interesting  and 
the  reason  of  the  practical  usefulness  of  these  numbers,  has 
not  the  least  bearing  on  the  nature  of  their  position  in  pure 
arithmetic  or  algebra ;  after  all  of  them  had  been  accepted 
as    numbers,   the    necessity   remained    of   justifying    this 

*  See  Gauss,  Complete  Works,  II,  p.  174. 


SYMBOLIC  CHARACTER   OF  ALGEBRA.  125 

acceptance  by  purely  algebraic  considerations.  This  was 
first  accomplished,  though  incompletely,  by  the  English, 
mathematician,  Peacock* 

Peacock  begins  with,  a  valuable  distinction  between  arith- 
metical and  symbolical  algebra.  Letters  are  employed  in 
the  former,  but  only  to  represent  positive  integers  and  frac- 
tions, subtraction  being  limited,  as  in  ordinary  arithmetic, 
to  the  case  where  subtrahend  is  less  than  minuend.  In  the 
latter,  on  the  other  hand,  the  symbols  are  left  altogether 
general,  untrammelled  at  the  outset  with  any  particular 
meanings  whatsoever. 

It  is  then  assumed  that  the  rules  of  operation  applying  to 
the  symbols  of  arithmetical  algebra  apply  without  altera- 
tion in  symbolical  algebra ;  the  meanings  of  the  operations 
themselves  and  their  results  being  derived  from  these  rules  of 
operation. 

This  assumption  Peacock  names  the  Principle  of  Perma- 
nence of  Equivalent  Forms,  and  illustrates  its  use  as  follows  :  t 

In  arithmetical  algebra,  when  a  >  b,  c  >  d,  it  may  readily 
be  demonstrated  that 

(a  —  b)  (c  —  d)  =  ac  —  ad  —  bc-{-  bd. 

By  the  principle  of  permanence,  it  follows  that 

(0  _  b)  (0  -  d)  =  0  x  0  -  0  x  d  -  b  x  0  +  bd, 
or  (-&)(-<*)  =  &<£ 

Or  again.  In  arithmetical  algebra  aman  =  am+n,  when  m 
and  n  are  positive  integers.  Applying  the  principle  of 
permanence,  p  p     p 

(aty  =  a?  -  a*  ...  to  q  factors 

-=  a|+|+ -t°  J  terms 

=  ap, 
whence  afl  —  i/ap. 

*  Arithmetical  and  Symbolical  Algebra,  1830  and  1845 ;  especially 
the  later  edition.     Also  British  Association  Reports,  1833. 
t  Algebra,  edition  of  1845,  §§  631,  569,  639. 


126  NUMBER-SYSTEM  OF  ALGEBRA. 

Here  the  meanings  of  the  product  (  —  b)  ( —  d)  and  of  the 
p 
symbol  aq  are  both  derived  from  certain  rules  of  operation 

in  arithmetical  algebra. 

Peacock  notices  that  the  symbol  =  also  has  a  wider  mean- 
ing in  symbolical  than  in  arithmetical  algebra ;  for  in  the 
former  =  means  that  "  the  expression  which  exists  on  one 
side  of  it  is  the  result  of  an  operation  which  is  indicated 
on  the  other  side  of  it  and  not  performed."  # 

He  also  points  out  that  the  terms  "real"  and  "imagi- 
nary" or  '-impossible"  are  relative,  depending  solely  on 
the  meanings  attaching  to  the  symbols  in  any  particular 
application  of  algebra.  For  a  quantity  is  real  when  it  can 
be  shown  to  correspond  to  any  real  or  possible  existence ; 
otherwise  it  is  imaginary. f  The  solution  of  the  problem :  to 
divide  a  group  of  5  men  into  3  equal  groups,  is  imaginary 
though  a  positive  fraction,  while  in  Argand's  geometry  the 
so-called  imaginary  is  real. 

The  principle  of  permanence  is  a  fine  statement  of  the 
assumption  on  which  the  reckoning  with  artificial  numbers 
depends,  and  the  statement  of  the  nature  of  this  dependence 
is  excellent.  Eegarded  as  an  attempt  at  a  complete  presen- 
tation of  the  doctrine  of  artificial  numbers,  however,  Pea- 
cock's Algebra  is  at  fault  in  classing  the  positive  fraction 
with  the  positive  integer  and  not  with  the  negative  and 
imaginary,  where  it  belongs,  in  ignoring  the  most  difficult 
of  all  artificial  numbers,  the  irrational,  in  not  defining  arti- 
ficial numbers  as  symbolic  results  of  operations,  but  princi- 
pally in  not  subjecting  the  operations  themselves  to  a  final 
analysis. 

126.  The  Fundamental  Laws  of  Algebra.  "Symbolical 
Algebras."  Of  the  fundamental  laws  to  which  this  analysis 
leads,  two,  the  commutative  and  distributive,  had  been 
noticed  years  before  Peacock  by  the  inventors  of  symbolic 

*  Algebra,  Appendix,  §  631.  t  Ibid.  §  557. 


SYMBOLIC  CHABACTER   OF  ALGEBBA.  127 

methods  in  the  differential  and  integral  calculus  as  being 
common  to  number  and  the  operation  of  differentiation.  In 
.  fact,  one  of  these  mathematicians,  Servois,*  introduced  the 
names  commutative  and  distributive. 

Moreover,  Peacock's  contemporary,  Gregory,  in  a  paper 
"On  the  Eeal  Nature  of  Symbolical  Algebra,"  which  ap- 
peared in  the  interim  between  the  two  editions  of  Peacock's 
Algebra,f  had  restated  these  two  laws,  and  had  made  their 
significance  very  clear. 

To  Gregory  the  formal  identity  of  complex  operations 
with  the  differential  operator  and  the  operations  of  numer- 
ical algebra  suggested  the  comprehensive  notion  of  algebra 
embodied  in  his  fine  definition :  "  symbolical  algebra  is  the 
science  which  treats  of  the  combination  of  operations  de- 
fined not  by  their  nature,  that  is,  by  what  they  are  or  what 
they  do,  but  by  the  laws  of  combination  to  which  they  are 
subject." 

This  definition  recognizes  the  possibility  of  an  entire  class 
of  algebras,  each  characterized  primarily  not  by  its  subject- 
matter,  but  by  its  operations  and  the  formed  laws  to  which  they 
are  subject;  and  in  which  the  algebra  of  the  complex  num- 
ber a  -f  ib  and  the  system  of  operations  with  the  differential 
operator  are  included,  the  two  (so  far  as  their  laws  are 
identical)  as  one  and  the  same  particular  case. 

So  long,  however,  as  no  "  algebras  "  existed  whose  laws 
differed  from  those  of  the  algebra  of  number,  this  definition 
had  only  a  speculative  value,  and  the  general  acceptance  of 

*  Gergonne's  Annates,  1813.  One  mnst  go  back  to  Enclid  for  the 
earliest  known  recognition  of  any  of  these  laws.  Euclid  demonstrated, 
of  integers  (Elements,  VII,  16),  that  ab=ba. 

t  In  1838.  See  The  Mathematical  Writings  of  D.  F.  Gregory,  p.  2. 
Among  other  writings  of  this  period,  which  promoted  a  correct  under- 
standing of  the  artificial  numbers,  should  be  mentioned  Gregory's 
interesting  paper,  "  On  a  Difficulty  in  the  Theory  of  Algebra,"'  Writ- 
ings, p.  235,  and  De  Morgan's  papers  "On  the  Foundation  of  Algebra " 
(1839,  1841 ;  Cambridge  Philosophical  Transactions,  VII). 


128  NUMBEB-SYSTEM  OF  ALGEBBA. 

the  dictum  that  the  laws  regulating  its  operations  consti- 
tuted the  essential  character  of  algebra  might  have  been 
long  delayed  had  not  Gregory's  paper  been  quickly  followed 
by  the  discovery  of  two  "algebras/'  the  quaternions  of 
Hamilton  and  the  Ausclelinungslelire  of  Grassmann,  in  which 
one  of  the  laws  of  the  algebra  of  number,  the  commutative 
law  for  multiplication,  had  lost  its  validity. 

127.  Quaternions.  According  to  his  own  account  of  the 
discovery,*  Hamilton  came  upon  quaternions  in  a  search 
for  a  second  imaginary  unit  to  correspond  to  the  perpendicu- 
lar which  may  be  drawn  in  space  to  the  lines  1  and  i. 

In  pursuance  of  this  idea  he  formed  the  expressions, 
a  +  ib  +jc,  x  +  iy  +jz,  in  which  a,  b,  c,  x,  y,  z  were  sup- 
posed to  be  real  numbers,  and  j  the  new  imaginary  unit 
sought,  and  set  their  product 

(a  +  ib  +jc)  (x  -f-  iy  -±-jz)  =  ax  —  by  —  cz-\-i  (ay  -J-  bx) 

+j  (az  +  ex)  +  ij  (bz  +  cy) . 

The  question  then  was,  what  interpretation  to  give  ij. 
It  would  not  do  to  set  it  equal  to  a' +  ibr -\- jc' ,  for  then  the 
theorem  that  the  modulus  of  a  product  is  equal  to  the 
product  of  the  moduli  of  its  factors,  which  it  seemed  indis- 
pensable to  maintain,  would  lose  its  validity ;  unless,  in- 
•deed,  a'  =  b'  =  c'  =  0,  and  therefore  ij  =  0,  a  very  unnatural 
supposition,  inasmuch  as  1  i  is  different  from  0. 

No  course  was  left  for  destroying  the  ij  term,  therefore, 
but  to  make  its  coefficient,  bz  -f-  cy,  vanish,  which  was  tanta- 
mount to  supposing,  since  b,  c,  y,  z  are  perfectly  general, 
that  ji  =  —  ij. 

Accepting  this  hypothesis,  denial  of  the  commutative  law 
as  it  was,  Hamilton  was  driven  to  the  conclusion  that  the 
system  upon  which  he  had  fallen  contained  at  least  three 
imaginary  units,  the  third  being  the  product  ij.     He  called 

*  Philosophical  Magazine,  II,  Vol.  25,  1844. 


SYMBOLIC  CHABACTEB   OF  ALGEBRA.  129 

this  ft,  took  as  general  complex  numbers  of  the  system, 
a  +  ib  +  jc  +  kd,  x  -\-iy  -\-jz  -\-kw,  quaternions,  built  their 
products,  and  assuming 

jk  =  —  kj  =  i 

ft£  =  —  ik=j, 

found  that  the  modulus  law  was  fulfilled. 

A  geometrical  interpretation  was  found  for  the  "imag- 
inary triplet"  ib  +Jc  -f-  kd,  by  making  its  coefficients,  b,  c,  d, 
the  rectangular  co-ordinates  of  a  point  in  space ;  the  line 
drawn  to  this  point  from  the  origin  picturing  the  triplet  by 
its  length  and  direction.  Such  directed  lines  Hamilton 
named  vectors. 

To  interpret  geometrically  the  multiplication  of  i  into  j, 
it  was  then  only  necessary  to  conceive  of  the  j  axis  as 
rigidly  connected  with  the  i  axis,  and  turned  by  it  through 
a  right  angle  in  the  jk  plane,  into  coincidence  with  the  k 
axis.  The  geometrical  meanings  of  other  operations  fol- 
lowed readily. 

In  a  second  paper,  published  in  the  same  volume  of  the 
Philosophical  Magazine,  Hamilton  compares  in  detail  the 
laws  of  operation  in  quaternions  and  the  algebra  of  number, 
for  the  first  time  explicitly  stating  and  naming  the  asso- 
ciative law. 

128.  Grassmann's  Ausdehnungslehre.  In  the  Ausdeh- 
nungslehre,  as  Grassmann  first  presented  it,  the  elementary 
magnitudes  are  vectors. 

The  fact  that  the  equation  AB  +  BC=  AC  always  holds 
among  the  segments  of  a  line,  when  account  is  taken  of 
their  directions  as  well  as  their  lengths,  suggested  the 
probable  usefulness  of  directed  lengths  in  general,  and  led 
Grassmann,  like  Argand,  to  make  trial  of  this  definition  of 


130  NUMBER-SYSTEM  OF  ALGEBRA. 

addition  for  the  general  case  of  three  points,  A,  B,  C,  not 
in  the  same  right  line. 

But  the  outcome  was  not  great  until  he  added  to  this  his 
definition  of  the  product  of  two  vectors.  He  took  as  the 
product  ab,  of  two  vectors,  a  and  &,the  parallelogram  gen- 
erated by  a  when  its  initial  point  is  carried  along  b  from 
initial  to  final  extremity. 

This  definition  makes  a  product  vanish  not  only  when 
one  of  the  vector  factors  vanishes,  but  also  when  the  two 
are  parallel.  It  clearly  conforms  to  the  distributive  law. 
On  the  other  hand,  since 

(a  +  b)  (a  -f-  b)  =  aa  -f  ab  +  ba  -f  bb, 
and  (a  +  b)  (a  +  b)  =  aa  =  bb  =  0, 

ab  -f-  ba  =  0,    or   ba  =  —  ab, 

the  commutative  law  for  multiplication  has  lost  its  validity, 
and,  as  in  quaternions,  an  interchange  of  factors  brings 
about  a  change  in  the  sign  of  the  product. 

The  opening  chapter  of  Grassmann's  first  treatise  on  the 
Ausdehnungslehre  (1844)  presents  with  admirable  clear- 
ness and  from  the  general  standpoint  of  what  he  calls 
" "Formenlehre "  (the  doctrine  of  forms),  the  fundamental 
laws  to  which  operations  are  subject  as  well  in  the  Aus- 
dehnungslehre as  in  common  algebra.  y 

129.  The  Doctrine  of  the  Artificial  Numbers  fully  Devel- 
oped. The  discovery  of  quaternions  and  the  Ausdehnungs- 
lehre made  the  algebra  of  number  in  reality  what  Gregory's 
definition  had  made  it  in  theory,  no  longer  the  sole  algebra, 
but  merely  one  of  a  class  of  algebras.  A  higher  standpoint 
was  created,  from  which  the  laws  of  this  algebra  could  be 
seen  in  proper  perspective.  Which  of  these  laws  were 
distinctive,  and  what  was  the  significance  of  each,  came 
out  clearly  enough  when  numerical  algebra  could  be  com- 
pared with  other  algebras  whose  characteristic  laws  were 
not  the  same  as  its  characteristic  laws. 


SYMBOLIC  CHARACTER    OF  ALGEBRA.  131 

The  doctrine  of  the  artificial  numbers  regarded  from 
this  point  of  view  —  as  symbolic  results  of  the  operations 
which  the  fundamental  laws  of  algebra  define  —  was  fully 
presented  for  the  negative,  fraction,  and  imaginary,  by 
Hankelf  in  his  Complexe  Zahlensystemen  (1867).  Hankel 
re-anhounced  Peacock's  principle  of  permanence.  The 
doctrine  of  the  irrational  now  accepted  by  mathematicians 
is  due  to  Weierstrass  and  G.  Cantor.* 

A  number  of  interesting  contributions  to  the  literature  of 
the  subject  have  been  made  recently ;  among  them  a  paper  f 
by  Kronecker  in  which  methods  are  proposed  for  avoiding 
the  artificial  numbers  by  the  use  of  congruences  and  "  inde- 
terminates,"  and  papers  t  by  Weierstrass,  Dedekind,  Holder, 
Study,  Scheffer,  and  Schur,  all  relating  to  the  theory  of 
general  complex  numbers  built  from  n  fundamental  units 
(see  page  40). 

*  See  Cantor  in  Mathematische  Annalen,  V,  p.  123,  XXI,  p.  567. 
The  first  paper  was  written  in  1871.  In  the  second,  Cantor  compares 
his  theory  with  that  of  Weierstrass,  and  also  with  the  theory  proposed 
by  Dedekind  in  his  Stetigkeit  und  irrationale  Zahlen  (1872). 

The  theory  of  the  irrational,  set  forth  in  Chapter  IV  of  the  first 
part  of  this  book,  is  Cantor's. 

t  Journal  fur  die  reine  und  angewandte  Mathematik,  Vol.  101, 
p.  337. 

|  Gottinger  Nachrichten  for  1884,  p.  395  ;  1885,  p.  141 ;  1889,  p.  34, 
p.  237.  Leipziger  Berichte  for  1889,  p.  177,  p.  290,  p.  400.  Mathe- 
mathische  Annalen,  XXXIII,  p.  49. 


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